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TRANSCRIPT
RD-Ai74 493 FUNDAMENTRL STUDIES IN FATIGUE AND FRACTURE MECHANICS i/1PHASE l(U) CRRNEGIE-HELLON UNIV PITTSBURGH PR DEPT OFMECHRNICRL ENGINEE.. G B SINCLRIR 07 OCT 96 SH-86-13
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q~j. FUNDAMENTAL STUDIES IN FATIGUE AND FRACTURE MECHANICS (Phase I)
Final Report on Air Force Office of Scientific Research
Contract No. AFOSR-85-0030 .. .,.
G. B. Sinclair
DTIC -"SM 86-13 NOV 2 6 September, 1986
-..r:;.,
Department of Mechanical EngineeringCarnegie Institute of Technology
Lii Carnegie-Mellon University -
,Pittsburgh, Pennsylvania 15213
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I DISCLAIMER NOTICE *THIS DOCUMENT IS BEST QUALITYPRACTICABLE. THE COPY FURNISHEDTO DTIC CONTAINED A SIGNIFICANTNUMBER OF PAGES WHICH DO NOTREPRODUCE LEGIBLY.
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GB. Sincl air 040016C66161 Air Force Office of Scientific Research
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Department of Mechanical Engineering Bolling Air Force BaseCarnegie-Mellon University Washington, D.C. 20332Pittsburgh, PA 15213 ____________________
Isf IA ME OFP UNDSPNIG/SPONSDRING OFF Ica SYMBOL W. PROCUREMENT INSTRUMENT IDENTIFICATION 011LBER0ORGANIIZAYIONl Air Force Office O *IeP~dk~
of Scientific Research N A AFOSR850030
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Bolling AFB. D.C. 20332Stde /j 232BSi..lf Cw~.U~eiFundamental Stde
in Fatigue and Fracture Mechanics (Phase I) _____________________
12, PERSONAL AIJTMO 0 (56. B. Sinclair12a TYPE OF REPORT 13b. TIME COVERED OaE OF REPORT (yr.. e.~j I.PG ONFinal Report FRO 1/15/85 0o4/14/86 1986, October 7 T 83
If SUPPLEMENTARY N4OTATION
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1S. ABSTRACT XoC..~ ome se ee of mtvwv M, mdpat 67 6h WO sern
The need to examine the very foundations of fractur mechanics became evident atthe close of an earlier AFOSR program [1]. In essence, he basic tenet of fracturemechanics is that the stress intensity factor, K, controls fracture and fatigue:here the former claim is critically examined. The underlying supporting arguments-the original energy argument of Griffith and the more modern K-controlled region view-are considered. These considerations demonstrate that there are questionable
assumptions in both, so that the viability of K as a damage parameter for fracturehas to be established by the physical evidence. The first question then is whetheror not the critical value of K, K c, is a material parameter: checking data showsit need not be. The second question is can the technology be usefully predictive,even in the most simple of situations: checking the data shows it to be unreliablein thi3 role. At this time then, it remains to ask similar questions,(cont'd)
20 DIST01IEUTION/AVAILASILITY Of ABSTRACT 21. ABSTRACT SECURITY CI.ASIPICATIOVN
UNCLASSIPIEO/UNVLIMITED SAME As R1PT 0 OTIC USERIS 0 Unclassified
George K. H-aritos, Major, USA NA eIfmeigde AV" Ce, - I
DD FORM 1473,83 APR EDITION OF I JAN 73 IS OBSOLETE. Unclassified40SECURiTy CLASSIFICAONO 5*C
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19. ABSTRACT (cont'd.)
concerning the role of K in fatigue crack growth and, certainly for the monotonicloading case, develop alternatives: these are the objectives of the second phaseof the program.
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FUNDAMENTAL STUDIES IN FATIGUE AND FRACTURE MECHANICS (PHASE I)
TABLE OF CONTENTS
' ."d
Summary ....... .... .............................. .
Introduction ...... ...... .. ............................. 2
Overview of results obtained in research program .... ... ............. 3 "
Concluding remarks .......... ... .......................... 10
Acknowledgements ... . . . . . . . . . . . . . . . . . . . . . . . . . 10
References ........ ..... ............................. 1
Appendices
1. On the singularities in Reissner's theory for the bending of elastic plates
2. Path independent integrals for computing stress intensity factors at sharp notches inelastic plates / A remark on the determination of Mode I and Mode II stress intensityfactors for sharp re-entrant corners
3. Some inherently unreliable practices in present day fracture mechanics / Furtherexamples of the unreliability of stress intensity estimates found via local fitting
4. On the capricious nature of elastic singul ,rities / The elastic analysis of three re-entrantcorners
5. On obtaining fracture toughness values from the literature
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,:, SUMMARY ,..\ "
The need to examine the very foundations of fracture mechanics became evident at
the close of an earlier AFOSR program [ 1]. In essence, the basic tenet of fracture - -
mechanics is that the stress intensity factor, K, controls fracture and fatigue: here the
former claim is critically examined. The underlying supporting arguments - the original
energy argument of Griffith and the more rtodern K-controlled region view - are k
considered. These considerations demonstrate that there are questionable assumptions in
both, so that the viability of K as a damage parameter for fracture has to be established ".-'-. .,.,..~
by the physical evidence. The first question then is whether or not the critical value of K,
Ki. is a material parameter, checking data shows it need not be. The second question is _,.._
.C.,. o
can the technology be usefully predictive, even in the most simple of situations: checking
the data shows it to be unreliable in this role. At this time then. it remains to ask similar
questions concerning the role of K in fatigue crack growth and, certainly for the monotonic
loading case, develop alternatives: these are the objectives of the second phase of the
program.
LZ
p."%
•0 . .
2
INTRODUCTION
In general, the primary objective of fracture mechanics is to predict when a
component will fail when the analysis of the associated configuration leads to singular
stresses. This is not an easy task, since the theory is saying that physically unsustainable
infinite stresses result from even infinitesimal loadings. At this time, the accepted approach ..
for dealing with such anomalous findings is Linear Elastic Fracture Mechanics (LEFM), at
least in the instance of cracked geometries. Essentially, LEFM selects the coefficient of
the stress singularity, the stress intensity factor, as the parameter which governs fracture
provided response is sufficiently brittle. As a result of an earlier AFOSR research program
[1], it became quite clear that this choice and the surrounding technology needed a full
examination. This report describes work undertaken in a one year study to this end.
In somewhat greater detail, the principal current activities within a fracture mechanics _
treatment of a monotonic loading situation may be classified as follows:
* Singularity recognition
* Evaluation of singularity participation
* Interpretation of the resulting quantified singular behaviour
We discuss findings with respect to each of these in turn in what follows.
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OVERVIEW OF RESULTS OBTAINED IN RESEARCH PROGRAM
* Singularity recognition
* Typically the presence and character of singularities is well understood -seldom does
the fracture mechanics community stumble on this all-important first step. This is[
especially so for the elastic analysis of geometries containing cracks. It is also usually the
case for the elastic treatment of other configurations, and for the handling of cracks within
* nonlinear theories. Asymptotic analyses continue to be contributed in these la*,ter instances,f
e.g., respectively, [2]* and some excellent articles in the recent conference [3]. These
last, incidentally, demonstrate that the nonphysical nature of singular stresses is retained .
when one entertains yielding. La., the problem of interpreting singularities is not
circumvented by admitting plastic flow. Basically this is because, physically, elastic response
must precede plastic and elastic analyses to date continue to be plagued by singularities.**
Singularity participation
In two dimensions, analytical solutions provide explicit and exact values of singularity
participation or stress intensity, K, in some situations. In practice. though. even two-
dimensional geometries are often too complex to be tractable to a purely analytical
approach Here, however, numerical analysis can be employed to good effect Principally.
* the finite element method, when used in conjunction with a path independent integral.
* furnishes a reliable and accurate means of determining K This is primarily because the finite
element method readily adapts to the various and diverse geometries encountered in
engineering, while path independent integrals are orthogonal to any other regular fields
present in the analysis of a particular configuration. Moreover, the approach is sufficiently
ef ficient so as to meet engineering accuracy requirements in return for modest
computational effort Such efficiency levels can be demonstrated on test problems with
Copy appended for convenience.
~There do exist elastic analyses of cracks which are apparently free from singularities -these, however,.l
either fail to load the crack tip or involve an integration/ finite length scale in effect and therefore do not reallyovercome the difficulty....
. .. .~ . ,~ - .. . . .
4
known exact solutions. For example, the path independent integrals devised in [4J* (5)* %
can be used to determine K to within 1% using fewer than 200 constant- strain-triangle j
elements. In all then, the development of the ability to establish singularity participation is
well in hand, at least in two-dimensional instances. And the extension of such methods to-
three dimensions is not seen as a further development of any great significance in the
overall question of the performance of fracture mechanics, [114
There are in use at present, nonetheless, some other methods of determining
singularity participation which can be shown to be quite unreliable. [7], [8] In
substance, these methods employ some kind of fit of local field data to infer K the basic
reasons f or the potontial unreliability of such local procedures are as follows. First, all
local procedures must consider quantities near but not at the crack tip. Second, at such
stations fields other than the key singular ones can contribute. Third, the extent of such
participation cannot be either completely controlled or fully accounted for. As a
consequence. for any given local procedure there exist problems on which it produces A
unacceptably erroneous results.
A corollary of the unreliability of local fitting procedures is that fracture criteria
based on some local field value cannot be reliably connected to the stress intensity
involved This means that criteria entailing crack opening displacement, crack opening angle.
or some so-called critical stress or strain at a nearby station, all fail in the second aspect
of fracture mechanics, namely establishing singularity participation. The successful
completion of this aspect is a prerequisite to successful interpretation. Accordingly it is
unreasonable to expect that any of today's local fracture criteria should prove of genuine
and significant value in practice.
Copy appended for convenience.
"Copy sent earler to AFOSR (final report an Contract No. AFOS-79-0078).
Singularity Interpretation
The first argument suggested in support of the singularity coefficient, or stress
intensity factor, as the controlling parameter in brittle fracture was, in effect, Griffith's
energy balance. In Griffith's approach, the energy source is the strain (potential) energy .. ,
released on fracture, while the sink is a surface energy term. Thus the controlling function
at the onset of fracture is the energy release rate, G. Irwin showed that G has a simple
relationship to K independent of geometry, so that Griffith's energy balance is equivalent to
the choice of the stress intensity factor as a damage parameter.
Griffith's work represented a major step forward at that time. Since, however, a
number of questions have been raised concerning the extent of its validity (e.g.. Goodier
[91). Moreover, if one simply applies the ideas to the fracture of a truly brittle material in
a unaxial tension test, one obtains [103* "
oro0 . (10
Here a, is the ultimate stress, t is the length of the specimen. Equation (1) stems fromU
the fact that the energy source has dimensions of F.L, where as the energy sink has
dimensions of F alone, F being force, L being length For the application to the tensile
test, L transpires to be t, the specimen length, While (1) is trendwise correct in tt
fracture stress decreases with increasing size, it is oversimplified. This is because C canU
also depend on other dimensions and f' s dependence on length need not be to the -Y2
exponent, or even particularly close to this precise relation. Indeed, for sufficiently large
specimens, au is generally regarded as a material properly, independent of geometry. Thus
this simple application of Griffith's ideas raises a question as to whether or not the
assumption of a surface energy term as the dominant energy source is really valid, or even
sufficiently accurate in certain circumstances. The answer, as far as cracked geometries
are concerned, has to lie with the physical evidence.
Copy ncluded in [1 ]
.1
-. .,. ... ;..-.--..-,.,,.. . .,.-..,, . .........-... ..- - .-......-........ . ..... ... . . .
The more modern argument suggested in support of the stress intensity as the .*
controlling factor in brittle fracture is as follows. For the K-field alone at the crack-tip.
the associated field quantities are clearly in violation of the underlying yet unpolicedr
assumptions of linear elasticity, viz, infinitesimal displacement gradients, stresses in theI A
elastic range, etc. However, some distance away from the crack tip these K-fields doj
comply with underlying assumptions and may be regarded as being valid. To fix ideas, call
the radius from the crack tip outside which the K-fields are valid r. Conversely, as one
approaches the crack tip for any configuration, there comes a point after which the K-
fields dominate any regular fields present Call this radius r. As a result, if r~ < rd an
annulus is formed which may reasonably be viewed as K controlled, i.e. a region in which K
sets the boundary conditions in effect for the process zone at the crack tip. Thus under
these circumstances, given similar fracture processes in the same material, K determines
when fracture occurs.
The drawback to this argument clearly is that r may not be less than r. Some-
indication that this can occur is contained in [11 12 El J, wherein three very similar pac--
man geometries under the same loading are treated. By changing the roof of the pac-
man's mouth slightly. K switches from on to off to on again. Here in some sense rd is .-
d.*,.,
going from some finite value, to zero, to some finite value. Hence at some point in this
sequence, no K-controlled annulus exists. And further examples can also be constructed.
Thus the question arises as to whether or not this approach is sufficiently reliable to be
useful in practice: again the answer rests with the physical evidence.
Turning to the physical evidence, perhaps the most basic question is to what extent is
the critical value of K, the fracture toughness, a material constant In particular, to what
degree is the plane strain fracture toughness, Kic a material constant, since this is probably
the most carefully governed by standards in measurement of all parameters in fracture A
mechanics today. A review for two materials which possibly have had the most extensive
Copy appended for convenience
-° . .. -- f r wq
7 V
%
r-. testing in this regard of all steel and aluminum alloys, respectively, is given in [13]' A ,..
summary of the results is contained in Table 1 (bars indicate mean values, A indicates 95%
confidence limit, ar is yield stress, K plane stress fracture toughness). In sum, K is
about five times more variable than the yield stress and accordingly quite dubious as a
material property.
Table 1. Critical value of K as a material property
Property ranges
Material Auy AKi AKY cc :::-:
7075 T6 Al +11% 0y +81% KIc +55% K -Ic- - C
4340 Steel +9% y +50% KIc
A second basic question in terms of physical evidence is how well can K be used to %
predict fracture in the simplest of situations. Focusing on geometrically similar
configurations subject to a change of scale alone - arguably the most fundamental of tests
of predictive capability - an extensive, if not comprehensive, survey of all the pertinent
data is furnished in [14] " [15]. The answer found is summarized in Fig 1 This
figure shows the percentage of times LEFM is within 10% of predicting fracture - useful
agreement - and the percentage of times it is within 5% - good agreement These
percentages are shown as a function of scaling with the number of independent tests
involved for each class of scale factors given in parentheses. When scaling is small (2-3)
there is really little to predict even so K fails to furnish useful predictions over 70% of
coDy appended for convenience
eCopy included in E l.
:.: .-...
V..
• .*- - < -"*- -l ', ':' ':%''.'" ."." %' ''.'':-':.':'d,.: '. .'. ."." .-.. ',.'. .', ,.. ,. , " .- ".. '., ,'" . .. , . . ..... ' . ".. . - . '.. .% . .' . , ,- .'. . '... . .-. -. : del '
% -. _'._.. .',. . ,. '.... ,"L .".",.'." _i . ,. . , '=. , . . . " _ ' ."- .. "",
80 % within±15% ofLEFM
70k70 - - -- % within ±10% of
LEFM
60
% 50 *4
40 I'
30I
20
I-- --- - -(5- (23
I0
1 2 3 4 5 6 7 8 9 2
Fig. 1. Predictive capability of K f or changes of scale alone.
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the time. When scaling is moderate (3-4), K fails to furnish useful results over 80% of
the time, while when it is significant (>4) it fares somewhat worse until eventually there are
no instances of satisfactory performance. And good predictive capability is even more
rare. This perf ormance in such carefully controlled laboratory situations could perhaps have
been expected given the outcome of the earlier question concerning Kc as a material
constant, but is nonetheless most unsatisfactory.
1.€ ,- -10
6" CONCLUDING REMARKS
Fracture mechanics today typically is successful in identifying and quantifying
singularities, prerequisites to a useful technology. The present interpretation of the
resulting quantified singularity, however, is not good enough. More precisely, the choice of
the stress intensity factor as the parameter governing brittle fracture under monotonic
loading, while appealing in its simplicity, is not up to handling the complex phenomena _"
involved. There is a serious and immediate need to improve our technology here.
Further investigations are warranted regarding the performance of fracture mechanics
in the fatigue or cyclic life arena Given the unsatisfactory performance of fracture
mechanics in the simpler monotonic loading instance, there would seem to be little reason
for optimism here. Probably significantly different alternatives will need to be developed
for both monotonic and cyclic loading before engineering can be provided with an adequate
technology.
Acknowledgements - We are pleased to acknowledge the continued valuable benefit of
discussions during the course of this work with J.W. Griffin of the Department ofP*1
Mechanical Engineering at Carnegie-Mellon University. The financial support of the Air
Force Office of Scientific Research is also appreciated. ,.
U 4:
11. . .""
REFERENCES
1. G.B. Sinclair, On the role of dimensionless elastic fracture mechanics. Report SM85-14, Dept of Mechanical Engineering, Carnegie-Mellon University (1985).
2. W.S. Burton and G.B. Sinclair, On the singularities in Reissner's theory for the bendingof elastic plates. Journal of Applied Mechanics, Vol. 53, pp. 220-222 (1986).
3. Tenth U.S. National Congress of Applied Mechanics, Abstracts of Papers. University ofTexas at Austin (1986).
4. GB. Sinclair, M. Okajima and J.H. Griffin, Path independent integrals for computing stressintensity factor at sharp notches in elastic plates. International Journal for NumericalMethods in Engineering, Vol. 20, pp. 999-1008 (1984).
5. G.B. Sinclair, A remark on the determination of Mode I and Mode II stress intensityfactors for sharp re-entrant corners. International Journal of Fracture, Vol. 27, pp.R81-R85 & Vol. 25, p. R 17 (1985).
6. W.S. Burton, GB. Sinclair. J.S. Solecki and J.L Swedlow, On the implications for LEFMof the three-dimensional aspects in some crack/surface intersection problems.International Journal of Fracture, Vol. 25, pp. 3-32 (1984).
7. GB. Sinclair, Some inherently unreliable practices in present day fracture mechanics._ ,,International Journal of Fracture, Vol. 28, pp. 3-16 (1985).
8. G.B. Sinclair, Further examples of the unreliability of stress intensity estimates found via- local fitting. Report SM 84-13, Department of Mechanical Engineering, Carnegie-Mellon
University (1984).
9. J.N. Goodier, Mathematical theory of equilibrium cracks. In Fracture (editorK Liebowitz), Vol. 2, Academic Press, New York (1968).
10. GB. Sinclair, Some comments on the Griffith-Irwin approach to fracture mechanics.Proceedings of the Nineteenth Midwestern Mechanics Conference, Columbus. Ohio, pp.139-140 (1985).
11. G.B. Sinclair and A.E. Chambers, On the capricious nature of elastic singularities. In [3),(1986).
12. A.E. Chambers and GB. Sinclair, The elastic analysis of three re-entrant corners.Report SM 86-10, Department of Mechanical Engineering, Carnegie-Mellon University(1986).
13. A.E. Chambers and GB. Sinclair, On obtaining fracture toughness values from theliterature. International Journal of Fracture, Vol. 30, pp. R 11-R15 (1986).
14. G.B. Sinclair and A.E. Chambers, Strength size effects and fracture mechanics: Whatdoes the physical evidence say? Engineering Fracture Mechanics, in press.
15. G.B. Sinclair and A.E Chambers, A bibliography of strength size effects for crackedspecimens. Report SM 85-10, Department of Mechanical Engineering, Carnegie-MellonUniversity 1984).
- " " - " - '--.' - -, ,.- .-'-& ",-:.> -- . .-.- .,- -- ... .... , . . .. .. ...° . . .-- ,... . . . ,. .. . .. .-i ;
-. - - ***~ t ~ & -.. .... rr-r~r ~F- r - -of
APPENDIX~~~~~~~~~~ 1:O*HJINUAIISIN.ISE'STER O
THE ENDIG OFELASIC PATE
A Brief Note is a short paper that presents a specific solution of technical interest in mechanics butwhich does not necessarily contain new general methods or results. A Brief Note should not exceed ICiJSO words or equivalent (a typical one-column figure or table is equivalent to 250 words; a one lineequation to 30 words). Brief Notes will be subject to the usual review procedures prior topublication. After approval such Notes will be published as soon as possible. The Notes should besubmitted to the Technical Editor of the JOURNAL or AP"LD MicANics. Discussions on the BriefNotes should be addressed to the Editorial Department, ASME, United Engineering Center. 345 p...East 47th Street, New York, N. Y. 10017, or to the Technical Editor of the JOUaNAL OF APPLIED line.MEiucKiCs. Discussions on Brief Notes appearing in this issue will be accepted until two months Poi: . ,
after publication. Readers who need more time to prepare a Discussion should request an extension tion | .'.
of the deadline from the Editorial Department, load "tion,
On the Singularities In Reissner's Theory O: /2for the lending of Elastic Plates -D(
w. S. Marto@' &d G. 3. SI edsr,
Wedge-shaped elastic plates under bending, with the edgesrforming the wedge vertex being either stress-free, clamped or fMg ,.simply supported, are characterized as to possible singularbehavior within the context ofRea0ner'splate theoy=0
IstroductiouProbably the first singularity analysis of an angular elastic
plate under bending is William's treatment using the clAsical --
theory 11]. In the classical theory it is possible to satisfy stress-free conditions at an edge solely in an approximate way, since .only two boundary conditions can be enforced and there are D .three stress resultants. As the boundary conditions play an im- on R, w "portant role in governing singular behavior at the vertex of mant re.w 'any comer in a plate it is to be expected that Reissner's theory defined(21, which admits three, physically-natural, boundary condi- ez"a/ !201-?tions on an edge, may offer an improved, albeit singular.representation in these instances. This is the expectation that FW1 I eSImewind WPAWS t Sm Mte equation
possibly motivated other analysts (e.g., Wang 13)) to perform i.e.,analyses of complete, individual, crack problems usingReissnet's theory rather than the classical, and indeed more 8
physically sensible results are derived in these analyses. shaped elastic plate under bending within the context of - .Specifically, for the crack-tip on the tensile face of the plate Reissner's theory. Next we establish suitable solution forms-the same hydrostatic singular field ahead of the crack as oc- for the dominant asymptotic response near the wedge vertex I
curs in the extensional case of a cracked plate under tension is and set down conditions for the existence of these fields. The r F ..found; this is in contrast to the classical bending theory in conditions basically involve the analysis of an eigenequation on f. He0 - .which the principal stresses ahead of the crack differ in sign for each pair of edge conditions considered. The note con- coordinat ."
and magnitude. As a result it would seem reasonable to at- dudes by displaying these eigenequations and discussing the On ea -_,S
tempt the analogue of Williams' study for the classical theory eigenvalues satisfying them which give rise to singularities. tions are t,II and explore the singularities in Reissner's plate theory for a combinedwider range of geometries than that investigated elsewhere and For luotlo.
for a full range of boundary conditions; this is the intent of the The plate has thickness A and occupies the open wedge folows: .
present note. region. R, Stress.
We begin by formulating a class of problems for a wedge- R= 1(r,0) 0<r<.m,-oa2<0<a/21(O<as2r), (I) Clamp ,
where (r,) are the polar coordinates of a point P in the wedge SimplylDupmn'fll of Manikw Ialii.i efr,, Cw"Mt-Mdo UUnivesity. with respect to the origin, 0, at the wedge vertex, and a is the onV=-,/
Pttibuhi. PA tS213. "-'.m, ,.- 19. vertex angle ( i ). distin c/Masmuomp tweed by ASMED (Fig, . distinct pro
9e6; IA rei vv July Is. tos. The plate is comprised of a homogeneous. isotropic, and apply on ea
220 1VoI. 53, MARCH 1986 Transactions of the ASME Journal of .
• .-', .--- "
BRIEF NOTES I
Table I Eigemeuatlons for Reissner's plate theory genseratingsingular moment resultants
Edge Conditions Eigenequation ConstantsStress-free/stress-free sin%a - CIA? C, - (-)*sina
'IClamped/clamped uinka - C2 C2 - I./)h2 * msn
Stress-free/simply spotd sin2)a - , ,-sin2aitClamped/simply supported sin2)a.a=CA C, -(- Il/a)sin2a
,'d Simply supported/simply coSXQ - C7 C, - (- )*Cosa4'.ICsupported%
.0Note: k- 1.2 for symmetric or anti-symmetric loading. respectively, and x (3- w)/(I +4 P). A-.
ef45linear elastic material having Young's modulus, E, and mnetric and anti-symmetric contributions. Thus, in effect, nineW Poisson's ratio, op. For this plate the stress resultants and rota- problems are considered. Ensuring bounded displacements
hs tions of Reissner's plate theory, in the absence of surface concludes our formulation and limits the singular beha~ioro- aditdwietifomltioofsthnstl not cmlti
loading, can be expressed in terms ofthe out or plane deflec- amtewhltisfrutonstentll c peettion, ae(r,9), and a single stress potential. 1&r,S). That is. suffices given our objective of characterizing possible singular
a, fields at the wedge vertex. We next consider the construction- - of suitable sets of asymptotic solutions.
ilb 5 C)Singularity AnalysisM ~ 00-~ r *rO0 From the form of the relations in (2). observe that if-
* - IC. OQ'5 1). X'(' t ). as r-O on R. X~ a constant, then the( .3w I aw W 82W~ shear resultants and rotations are 0(rt ), while the moment
- ,a 7-- r---r--+-iOF resultants are 0(r11-). Accordingly we seek to construct-D 2w P aw V 02W separable solutions for w, X satisfying the governing equations
.4f,, 2-9)- (3) which furnish six independent constants multiplying these891 aF Br .2 : / dominant contributions as r-0. thereby providing a means of
satisfying the six boundary conditions contained in any pair of1 &2 Ox(2) the edge conditions (4). To this end, and noting that X -X
Mo -f 802 rr and DV 2w are harmonic functions and the interrelations in(3), we therefore take as our asymptotic solution forms for X
1=0D--(+~) and w, the bihaionic unction
2-y' OX I aw .Dol-) Or 7 -9-' as r-O on R,where
2-v 1Ox w F~t*) -(bcosQ,+ 00 + b2sin~s-i+l06D( )r 89 *r *bscosQ.\- l)#+b,$sinQX - 1)0),
on R, where V, Vv, the shear resultants, Alt, M, Ma, the MO- G(o,.)=(bsginQo4 1)0+ bacos~s-- 00)9'%'. ment resultants and 00, 0,, the rotations, are functions of r, 9, .%.
defined in the usual manner, with -h 2 /iO1. D-Eh'/-bsn- )+yloss ))D~ 2 12(1 - P3). the last being the flexural rigidity. Then the field The stress and moment resultants and the rotations in (2) may
equations of Reissner's theory are reduced to the Cauchy- then be written as"aRiemann equations for the function$ %-.yV2 X and DV~w. V,-i. r 0Q+ ~ -2). , (A~ - + l)i tF+ 0(r-*2),
text of r~Y x ~ -- D ~) ~r'2,'-(~-:.frs(3) M'('rAJy(F'Qs -(+ X- )-D( -P))O'1+0(r* ),
- (-'r 2 ~ -(DV 2 w), 2___~~~~tex~~' + a aD.r~ ~ I)F G'40r.) (6)-Ids.The r+0r
equa*tionl on R. Here V 2 is the L.aplacian operator in cylindrical Polar . '-
snthe n coneachted e 2-yi.0$snthe n coordinaes, e face three homogeneous boundary condi- 19, D.)F- P
,arittes. tions are to be satisfied. These three boundary conditions arecombined in sets of edge conditions to model various edges as aspect0ton 0.weetepimsdnt ifeetain t
V ~~follows:mecto9Cis~ wedge With this set of separable functions for w and X the
Stress-free Al.# - Me, -0, V@ - 0. singularity analysis proceeds in a manner similar to that
,). (a, ' ~~~~~~~~~~~~~Dempsey and Sinclair 151 for logarithmic singularities. Impos- ri~~ -- 4 eeoe yWlim 4 o oe iglrte n y,, ,!
the wedge Simply supported MO -0,00. WOO0, ing the displacement regularity requirements on (6) and con-a a is the on 0' */u2 (Os r<ft). These three cases combine to give six
distinct problems for the wedge. When the same conditions =Nr ha am cano m solution 121 ad have a tia~wic,en.,ropic, and apply on each face, it is possible to distinguish between gym- mamiber of id~mu consants availabse for ahe aympmKc auaa~va .
*AW ASME Journal of Applied Meachanics MARCH 1986, Vol. 531221
Jp
* ~ L' - - --. - 7. 7 1 V, -
BRIEF NOTES ,
fining attention to singular solutions, the resulting conditions for the first three cases in Table I by Williams 141 and are charmay be summarized by decomposed in effect into symmetric and anti-symmetric pas fecN - 0(rl -1) for real X satisfying D -0, 0 <X < 1, by Kalandiia 161, thus accounting for the first five cases. The the
roots for the last case, simply supported/simply supported, amsin(elnr) can be determined by inspection. Some of the eigenvalues
1) for complex X + iif given in [4, 61 are actually the real parts of complex solutions; E41ficos(VIr) however, no truly comprehensive search for complex roots ap-
pears to be available in the literature. Such a parameter study . u ,satisfying D -, 0 < AReX < 1, (7) is outside the scope of the present work. Nonetheless, for any dir .,
given application the determination of complex eigenvalues t io ,proceeds routinely on separating the pertinent eigenequation a -M-0(i-k- 11mr) for real A satisfying for the specific ct-value into real and imaginary parts and solv- __ - i
d-OD ing the resulting, simple, simultaneous pair of transcendental,0., m<n, 0< k< I, equations. Likewise, logarithmic singularities have not beenexhaustively searched for, but are straight forward to check
sin(sinr) for in any specific instance. w ,
M=0( -'mrf -or complex X = +in In conclusion we remark that the correspondence between z _cos(rtlnr) the singular fields in Reissner's theory and those in extensional
plate theory is not restricted merely to the singular eigen-tr-OD values, but carries over to the actual eigenfunctions which 2
satisfying =, m < n, 0 < ReX s 1, share the same r and 0 dependences, as can be deduced from athe solution (6) and its counterpart in Williams [41. E,
as r-O on R, where M = (M@, t, M, is the vector of mo- Refnceolment resultants. In (7), D is the determinant of the coefficient I Williams, M. L.. "Surface Stress Singularities Resuing From Various titm atrix stem m ilg from the substitution o f (6) into a set o f ed ge llo vnsj Coof ~i .S An INAMW Co r es of Ap l ifi Ud e Blii. " Pr9 -1.pp
conditions drawn from (4) and n is the order of this matrix, m 35-329 ,A* . I.mits rank. For any particular combination of the edge condi- 2 ReisI,.. E.. "The E 'ec of Tramr, I Sha Defrmio on itt. .e-dintions (4) for a wedge angle a, the values of X in the ranges of Elastic Plats." ASME Joua AL os ArtuD hMacuArncs, Vol. 12,1 945. pp.given in (7) may be regarded as the singular etsenvalues of the A-9A77. y
3 Waneg, N-M,, "Effects of Plate Tic-knes on the Se"nn of an Elastic' - '''
eigenequation, D = O. We now investigate the eigenequations lMe Contanng a Circ." ,curna/of tet msd Phs,. Vol. 47, 1968.eullins from such expansions. pr 37-)Xo b -4 Wil:kurns. M. L.. "-Stress Singulanitses Resulting from Variou Boundary .""'
J [tnL~quIJ(JOOSCondmloi.. :n Angular' Cornes of Plates in Extsion.," ASME JOviL. .L or Ar,-
,tiID MICK&MICs. Vol 19. 1952, pp. 526-528.5 Dempsey. J P.. and Sinclair, G. B.. -O tbe Stress Singularities in the
For the particular problem of the symmetric bending of a e o ,o w.." , Emev. Vol.9, t9 Istress-free/stress-free wedge, substituting (6) with Pl. o mi)9l
b, - b, = b, - 0 therein into the first of (4) and expanding the 6 Kajandit., A. L.. "'Remarks on the Singularity of Elastic Solutions Near fdeterminant of the resulting 3 x 3 coefficient matrix leads to Co.eTn," Prt t ed MEUtAd A MAhmie, Vol. 33, 199. pp. 232-135.
sin(X+ l)a/2(ksina + sin)a) -0(0< a s 2r). (8) %
Equation (8) factors into two equations; however, the first of 'these, while not generating a completely trivial solution, does
%
not give rise to any moment resultants and therefore con- Dynamic Blehlvior of Beam Structures 2.tributes no singular fields. Consequently, it may be discarded Carrying Moving Maasileaving as our eigenequation for this case only the second fac-tor. The eigenequations for the remaining combinations of the S- SWIgI'edge conditions each possess similar, non-singular, ItUdectioi -multiplicative factors. in Table 1, we suppress these and listonly those parts of each eilgenequation that have attendant The dynamic response of structures carrying moving masses A'singular fields, is a problem of widespread practical significance. A detailed , .
Comparison of the first three cases in Table I with the cor. survey of research efforts in this field was given by Stanisic: etresponding extensional cases given in Williams 141 shows the a. 121. The original problem is nonlinear in both local and Neigenequations to be identical. Examining the fourth and fifth convective derivatives 131 and is complicated by the presencecases in Table I and noting that the conditions for the simply of a Dirac-Delt% function as a coefficient in the differentialsupported edge in (4) are the same as anti-symmetry re- equation of motion. Previous methods [21 applied for thequirements, we see these eigenequations are equivalent to the solution of this problem are approximate in nature andanti-symmetric pans of the first and second eilgenequations, tedious in their hierarchy of mathematical operation. Recent-respectively, for a wedge of angle 2a. It follows that these two ly, Staniic (31 expressed the solution in terms of eigenfunc-cases are also effectively contained in Williams' extensional tions satisfying the boundary, initial and transient conditions,analysis 14). Finally, taking as the physical analogue of the for a heavy mass moving over a simply supported beam.simply supported/simply supported edge condition, the exten- However, in engineering practice there are problems that in-sional anti-symmetry conditions, u, - 0, og = 0 where u, is the volve more complex boundary conditions and, therefore, it isradial displacement and o, is the tangential normal stress, we of phenomenological interest to look into the physics of thefind that the last case too has a corresponding extensional dynamical behavior of a clamped and a cantilever beam under ,i
eigenequation. The significance of this correspondence is that the action of heavy moving masses. The present study extendsdiscussions in the literature on the extensional eigenequations Stanisit's theory 131 to study the dynamic behavior of a
are directly applicable to elastic wedges generated by -'.'.-
R bending theory. s1chool of Aeronautics and Astonaucs. Purdue Univitty. WeilSolutions for the dominant singular real pan of X are given Lafay ,. Indiana 47w. Stude ,t Mmnsur. ASME.
2221 Vol. 53, MARCH 1986 Transactiona of the ASME .-
LA~ .~ ~A~ A x''A..2",h
;4.*.%
APPEDIX : PTH IDEPNDEN INTGRAS FO COPUTIG STESSINTESIT
FACTORS ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ .. ATSAPNTHE NEATC..TS/AREAKO H
DETERMINATION ~ ~ ~ ~ ~ ~ ~ ~ ~. OF MOEIADMD 1-TESITNIY.ATR O HR
RE-ENTRAT CORNER
%44*
INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING. VOL. 20. 999-1008 (1984)
PATH INDEPENDENT INTEGRALS FOR COMPUTINGSTRESS INTENSITY FACTORS AT SHARP NOTCHES
IN ELASTIC PLATES
G. B SINCLAIR. M OKAJIMA AND J H GRIFFIN
Department of Mechanical Engineering and the Center for the Joining of Materials, Carnegie-Mellon University,Pittsburgh. Pennsylvania. U.S.A.
SUMMARY
A set of path independent integrals is constructed for the calculation of the generalized stress intensityfactors occurring in elastic plates having sharp re-entrant corners or notches with stress-free faces andsubjected to Mode 1, 11 or IIl type loading. The Mode I integral is then demonstrated to enjoy a reasonabledegree of numerical path independence in a finite element analysis of a test problem having an exactsolution. Finally, this integral is used on the same problem in conjunction with a regularizing, finiteelement, procedure or superposition method. The results indicate that sufficiently accurate estimates ofthese stress intensity factors for engineering purposes can be achieved with little computational effort.
INTRODUCTION
In a number of engineering structures, sharp re-entrant comners or notches are introduced,usually to facilitate fabrication. At the vertices of these notches, stress concentrations arisemaking them likely sites for fatigue crack initiation and therefore the potential sources ofultimate failure. Elastic analyses of such configurations result in stress singularities (see, e.g., :References I and 2). While these singularities are physically unrealistic in themselves, it ispossible that, with a sufficiently accurate determination of their nature and participation, anextension of the now accepted notion of a stress intensity factor K might prove of value.Although such a generalization lacks the thermodynamic argument in terms of energy releaserates that underlies the physical significance of K for cracks, it could lead to an improvementover present practice in initiation calculations for notches. One current approach to suchcalculations is to assume that a crack has actually formed at the notch tip, then compute theusual stress intensity factor variation under cyclic loading, AK, and compare it with materialthreshold values to see if crack growth commences: with a generalized K for the notch itself,the AK could be computed directly and compared with an accompanying set of experimentalthreshold values for the notch. In this way, increased life could be detected in circumstancesin which load levels are too low for the formation of a crack at the notch tip. The intent ofthis investigation is to contribute to the analytical component of such a technology by developingan efficient computational procedure for accurately assessing generalized stress intensity factorsat stress-free notches in elastic plates, one which is readily applicable to the wide range ofcomplex configurations encountered in practice.
The basis of our approach to meet the foregoing objective lies in the development of a setof path independent integrals which pick off the K at notches. Such path independent integralsfor singular elasticity problems probably owe their origin to Eshelby's work 3 in the early 1950s,and a number of different forms for these integrals have evolved over the intervening years
0029-5981 /84/060999-OSOl .00 Received May 1982@ 1984 by John Wiley & Sons, Ltd. Revised 20 March 1983
6-
7. 7.'7. 7,-;'.-
1000 G B. SINCLAIR. M. OKAJIMA AND J. H GRIFFIN
(e.g. Sanders* integral; 4 see also Freund' and the references contained therein). All of theforegoing integrals apply to a crack. Recently Stern and Soni 6 developed an integral for aright-angle corner under fixed-free conditions; here we follow the ideas in Reference 6 andconstruct a set of like integrals, termed H integrals, for a stress-free notch in an elastic plate.
The computational advantage of path independent integrals in the analysis of singularelasticity problems stems from the fact that the only numerical errors accrued in their calculation F51
derive from the numerical approximations being used-there is no additional source of errorssuch as that due to the truncation of an eigenfunction expansion as in local fitting methods. Itis to be expected therefore that, when used in conjunction with a finite element method whichrecognizes and handles the singularity present or some equivalent numerical approach, aprocedure results which is easily adapted to varying configurations yet which yields the necessaryresolution for practical purposes in return for a modest computational effort. Such is indeed - -
demonstrated to be the case in the second section of the paper, wherein the numerical pathindependence of an H integral drawn from the previous section is examined, following which "*"- ,the same integral is used in conjunction with the regularizing procedure of Sinclair and Mullan."The paper then closes with some remarks on other applications and extensions.
CONSTRUCTION OF THE PATH INDEPENDENT INTEGRALS
Here we first focus on the in-plane symmetric loading of a notch in an elastic plate and developa path independent integral for the stress intensity factors in this class of problems. We thendiscuss other integrals for anti-symmetric and out-of-plane configurations.
To formulate our initial class of symmetric, in-plane, notch problems we consider a planefinite wedge subtending an angle 2a at its vertex (r/2 <a < ir), let (x,, x 2) be rectangularcartesian co-ordinates having origin at the wedge vertex, the x,-axis bisecting the angle there,and take R to be the open region defined by that part of the wedge in the upper half-plane, . -
with R being its boundary (Figure 1). To facilitate further the formulation and subsequent
X2.
Figure 1. Geometry and co-ordints for the region R
2.- .- %-
*...- .
r V
PATH INDEPENDENT INTEGRALS 1001..' .,*
analysis. we introduce companion cylindrical polar co-ordinates (r, 0) related to (xi, x2) by thetransformations, "" "
x, f=rcos0 x2 =rsin 0 (Or <cc, -,r< 0 ) (1)
Then, in general. we seek the stresses or, and displacements u, (i,j= 1,2) throughout R asfunctions of x, x, satisfying the following. The two-dimensional stress equations of equilibrium ,.in the absence of body forces,t
0,, =o on R (2)
The stress-displacement relations for a homogeneous and isotropic. linear elastic plate.
, :[((3- c)/(w - 1))5,,U 5 + U,,+ u,,] on R (3)
wherein , is the shear modulus, K equals 3-4t, for plane strain and (3-t,)/(1+ ,,) for planestress t, being Poisson's ratio, and 6,, is the Kronecker delta. The stress-free conditions on thenotch face,
o,.= 0 on aR when XI/X2-cot C (4)
where n, are the components of the unit outward normal to RR. The symmetry requirementsahead of the notch tip.
47120 u 2 =0 onrRwhenx 2=0 (5)
The boundary conditions prescribing the tractions or displacements on the remainder of 8R,
a',fn,f= s? ona ,R u, - u? on a2R (6)
wherein s,, u, are the applied tractions and displacements and atR, a 2R are complementarysubsets of i)R excluding the intervals in (4), (5). And finally the regularity requirements at thenotch-tip which insist that the displacements are bounded there,
u,0() onRasr-O (7)
Specifically we seek to extract from the stress and displacement fields meeting these require-ments the generalized stress intensity factor present defined by
K, - lim V(21r)x'0 2 onx 2 0 (8)
Here the subscript I denotes the opening symmetric mode, Mode 1, and A is the singulareigenvalue characterizing the only singular stress field possible at the re-entrant corner asidentified after Williams.2 t That is, A is the root of the transcendental eigenequation,
sin 2Aain-A sin 2& (0< A <1) (9)
with an associated eigenfunction having stresses and displacements in the neighbourhood ofthe notch tip behaving in accordance with
o,,= O(rA-1) u,-O(r') onRasr-'0 (10)
1 Although the lower case subscripts only range over the integers (1,2), the usual index notation conventions apply:repeated subscripts imply summation and a subscript preceded by a comma indicates partial differentiation with ,,..
respect to the corresponding cartesian co-ordinate. -p.'I See Gregor)' for a completeness argument for Williams' eigenfunctions.
1' ... o
• ' y: -'- ~. ~ . - . . -
Irr
Elastica plate. wihnoc
1002 0. B SINCLAIR. M OKAJIMA AND J. H4. GRIFFIN '',
.. " %
,..5
rFigure 2. H, integral paths
To construct a path independent integral for computing KI we proceed as follows. In general,
path independent integrals can be devised by invoking the divergence theorem of Gauss on
ensuring the divergence of the integrand is zero: in elasticity, this can be done in effect withBetti's reciprocal theorem (refer, e.g., to Reference 9, p. 355) which, in a plane, can be stated "
.as* -0 )
contou inun dIn (11), the integration is performed in a counter-clockwise sense around r which is any closedcontour in R, a*, u, are complementary fields satisfying the same field equations as cr, u,, -
namely (2), (3), n, are now the components of the unit outward normal to F, and ds is aninfinitesimal line segment of r. Next we choose r as the contour comprised of any path Iwhich emanates on the lower notch flank and terminates on the upper, a circular path of radius
r from the upper flank to the lower, and two closing straight paths along the flanks (Figure
2). On letting the radius of the inner circular arc shrink to zero it is clear that only those parts
of the integrand which behave like 0(0/r) as r-.0 can contribute to this portion of the integral"
in (11). In order that such contributions stem from the singular K, fields alone we therefore .¢
require that*, -O(r--) 0 =O(r') onRasr--0 (12)
Then (10), (12) imply that the product of the K, terms with the complementary field in (11)
has the desired l/r behaviour while all other fields in cr, u, contribute terms o(1/r) as r-0.
by virtue of the fact that the K, singularity is the dominant one present. Unfortunately, under
this limiting procedure the possibility of divergent integrals along the flanks arises because of
the potential of I/r terms there. Fortunately, if A is an eigenvalue satisfying (9) so is -A, so
that the complementary fields can be further required to be eigenfunctions themselves, thereby
rendering both c,,n, and ar~n, zero on the flanks and making the integrals there identically
zero. Under these conditions the integral around I equals the counter-clockwise integral around
00 the circular arc in the limit as r-. 0 which, with the selection of appropriate participation factor
-"-. .
v-i 7 -7
PATH INDEPENDENT INTEGRALS 1003
in the complementary eigenfunction, can be adjusted to recover K1. That is, we define H, by
and scale the complementary cigenfunction satisfying (2)-(5), (12) so that
H, = K,(14)The actual forms for the complementary eigenfunction are most readily established in polarco-ordinates and can be derived along the lines of Williams' analss yielding
c =-K~r-AI[(A +3) cos (A + 1)6-p cos (A - 1)0]
a,,=K r-A -'(A - 1) cos (A + 1)0-0 cos (A - 1)0]
a*,. -K 1*r-'AI(A + 1) sin (A + 1) 6-,6 sin (A - 1) 6) (15)
u? =[(A + K) cos (A + 1)0-0 cos (A - 1)0]20AA
withN/(8 7r) 1A(A sin2 a +sin A a) A2-(I1+ o)(sin 2a +2a cos 2Aa) A cos 2a+cos 2Aa
r Before moving on to the application of the H, integral of (13), several remarks concerningits use and some simple extensions are in order. First, because of the symmetry involved, it isclear that H, can be computed as twice the integral around that part of I within R alone.Second, (14) holds in the limit a - 7Y and the notch becomes a crack. Third, the pure tractionconditions on iR and the pure displacement conditions on a2R are solely for convenience inthe formulation and can be relaxed to accommodate admissible mixtures of traction anddisplacement components. Fourth, extension to the antisymmetric Or Mode 11 case is straighitfor-ward and leads to
H11 - K1, J (a',,u*'-ff*u, )n,ds (16)
where K1, is the Mode 11, stress intensity factor defined as the natural generalization to a notchof the Mode 11 factor for a crack, the as,, u, components now fulfil antisymmetry requirementsinstead of symmetry, i.e.
OL022in0 u1 -0 onARwhenX2-0 (17)
and, in polar co-ordinates, the antisymmetric complementary fields are given by:a'?.= - Kl~rA'-'[,8sin (A - 1)0 -(A + 3) sin (A + 1)0]
a*". mK 11r A -I p sin (A -1)0--(A-1) sin (A + 1)0)
a*,*. =K JrAI cos (A -1)0-(A +1) cos (A +1)8] (18)-- A
u:-- 2$'cO'c A- )-(A - )cos (A +1)0]2psA
.. .-. •
.% 4..%'.w
1004 G. B SINCLAIR, M. OKAJIMA AND J. H, GRIFFIN
with
' M/(8ir)MA(A sin' a-sin" A ) A2-1
(I +x)(sin2a-2acos2Aa) Acos2a-cos2Aa
where A is now the root of the antisymmetric eigenvalue equation, namely (9) with the minussign changed to plus. Fifth, for problems entailing a mixture of Modes I and 11, the fields necd ..-
to be separated into their respective symmetric and antisymmetric parts in accordance with -'..
,*4 -[u,(x, x2)r (-)'u, (x,, -x2 )/2 (19)
where u, and the stresses derived from substituting u7 into (3) constitute the symmetric part,u,. etc., the antisymmetric, then treated individually using (13) and (16), respectively. Sixthand last, extension to include the outstanding ouw-of-plane mode, Mode 1II, also proceedsroutinely and results in
Hill = KillJ (0'3,u* - 17,u 3 ) ds, (20)
where Kil is the generalized, Mode 111, stress intensity factor for a notch, the of3/, U3 componentsare now the out-of-plane responses ( 1I, 2), and, in polar co-ordinates (x 3 - Z), the out-of-plane complementary fields are:
-U2 (2 ). I 2 sin-2.,) (27"-T)
Cos K,-- 2%/(2r)r+W/2 " 2a 2a.
TEST PROBLEM
In this section a plane elastic notch problem having an exact solution is set up and analysedusing a finite element approach and the HI integral. The results found enable the numericalpath independence of the integral to be examined and its efficiency as a computational deviceassessed.
For simplicity we fix attention on a problem drawn from the restricted class formulated inthe previous section. One means of setting up a problem within this class having an exactsolution is to superimpose symmetric eigenfunctions for a notch on some region R, and weadopt this simple device here. In order to thoroughly evaluate the performance of the integralwhen used with the simplest of finite elements, the constant-stress triangle. we select forcombination the first three eigenfunctions for the notch since these contain: the necessarysingular stress field; a stress field which, while continuous, is not continuously differentiableand may therefore be regarded as just barely being a regular field for constant-stress elements;and, lastly, a stress field which is relatively smooth and is continuously diflerentiable.t Moreprecisely, we consider a plate with a 90* re-entrant corner (a - 3v/4) and take R to be thetrapezium within the upper half-plate (Figure 3) defined by
SR X21,, )-X 2 < x, < a 0 < X < a) (22) %
O Dy a 'regular' field with respect to a gien element we mean one which does not impair the normal maximum %
convergence the element can enjoy; thus fields are regular for analysis using a constant-stress element it the stresses %. ,-
are continuous (ec Reference 10, Section 2.2. for a full discussion of the regularity requirements for vanous finite * ".elements).
.... ... ... .... ... ... ...
yL .N " ~~x wY W Y. v~~ ~I ~~' *.* - -
4, .- o 4-.
PATH INDEPENDENT INTEGRALS 1005
.4
.4 21I 12 13 E.4:
Figure 3. Geometry for the test problem, coarse grid and integratiom paths.
On this region we are to place the first three symmetric eigenfunctions for the 90* notch.admissible in the sense of (7), and associated with the first three eigenvalues of (9) witha - 3 r/4 there and having Re A >0. The forms for superposition are therefore those of (5)with K * therein being replaced by the constants K,, C 2, C3 and A being exchanged for -A
then set equal to the three corresponding eigenvalues, As, A2 , A3, provided the real pans ofthe expressions in (15) are taken in the event of a complex eigenvalue. In the interests of 4brevity we suppress details of these fields here and merely list the required eigenvaluesdetermined numerically from (9):
A - 0.544 A2 - 1.629+ i 0.231 A3 - 2.972+ i 0.374 (23)
The corresponding eigenfunctions satisfy the two-dimensional field equations of elasticity (2),(3). the stress-free boundary conditions on the upper notch face (4), and the symmetry *
-4 conditions ahead of the crack (5). In combining them we first adjust the participation of theAI-fields so that
K,/o°%(2v)a' - ' -1 (24)
Here K, is the generalized stress intensity factor defined in (8) and cr° equals all for A, at.. , - a. z2 0. Then, to complete our test problem, we add together all three eigenfunctionswith the A2-, A3-fields having participation factors C., C 3 such that they share the same l, Imagnitude at X, -a, X2-0 as the A,-field (namely 0o) and take, as the boundary conditionson the remainder of aR (XI a, 0 < Xz < a; - a < x, < a, x2 - a), the tractions that result therefrom their superposition. In this manner all three fields contribute to a comparable degree to
A the test problem.The application of a constant-stress triangle, finite element code and the calculation of the p q
* . H, integral for this test problem proceeds routinely. Since the issue of interest is the computa-tional performance of the integral, we discretize R uniformly with a set of 45" sosceles triangles.
"* To examine convergence we employ a sequence of three grids-a coarse, a medium and a .." fine-with the medium and fine grids being formed from the coarse by successively halving*', the element sides. The coarse grid has 48 elements and 35 nodes (Figure 3), the medium 192 .4
elements and 117 nodes, and the fine 768 elements and 425 nodes. To examine path inde-pendence we use three contours Y. (n - 1,2, 3) which have nodes common to all three grids
4. 0.-4".
.-4, .
• -"4
.'+. ""..", ..'"....-\.. . .- ' ..""' . . : "" ." - . ' -* .-. ","'.""""4 *""' ' " , -, " - "-". - -
-.. .. . . . . . . ..... . . ..+ ; .+: - ; . ' ,* .~.:.. , ... ',,V.~ W.° -7-.
1006 G. .SINCLAIR. M. OKAJIMA AND J. H. GRIFFIN , -
(Figure 3) and along these contours use the mid-point rule to evaluate the integral in (13)-aquadrature rule of like accuracy to that of the finite element discretization. The results sofound are presented in the first half of Table I. .,
.- -.
Table 1. Numerical values of the dimensionless stress intensity factor for the test problem (exact value 1)
Direct calculation using (13) Calculation using regularizing - ,on three integration paths procedure (25) on three paths
Grid 7, 12 . 1 12 1 -
Coarse 1.044 0.843 0.883 0.979 0.975 0.978Medium 0.913 0.939 0.940 0.992 0.994 0.994Fine 0.972 0.973 0-972 0998 0.998 0.999
Of course the problem at hand is singular and accordingly results from the foregoing directtreatment can be expected to be poor; needed is an approach which recognizes the singularitypresent and takes it into account if better results are to be realized. One procedure with thisattribute is described by Sinclair and Mullan' and belongs to a class of auperposition methodswhich, in essence, treat the singular part of any problem analytically with scant regard forboundary conditions remote from the singularity source, then use a simple numerical methodto complete the satisfaction of the boundary conditions in what is now a regular problem.What basically distinguishes the procedure in Reference 7 from other superposition methodsis the use of path independent integrals to balance the analytical and numerical contributionsand it thus seems natural to apply it here. The end result so far as the stress intensity factoris concerned is (see Reference 7 for details of the surrounding argument) 5
K3 -b A,/A;ils (25)
wherein R, is the numerical estimate of H, for the problem of interest, R1 the numericalestimate of Hj for an associated singular problem formed by taking corresponding boundaryconditions on OR as prescribed in effect by the singular eigeniunction alone with a participationfactor of unity. What (25) says, in effect, is not that either A, or /4 represent accurate estimatesof HI or Hl, respectively, but rather that they constitute roughly equally crude estimates withthe errors cancelling in large part. Central to this cancellation is the use of an underlying --
method for calculating K, which has predictable errors due solely to the discretization--hencethe need for the path independent integrals in (25).t
Application of (25) to the test problem is straightforward: the Ri values that have alreadybeen computed are simply divided by the numerical values of H, found on the same grid andpath for the singular problem in which the boundary conditions on OR are established by theplacing of the A,-fields alone on R with a dimensionless participation factor of unity, i.e. as -in (24). The results so found are presented in the second half of Table I
The numbers in Table I illustrate the degree to which the H, integral is numerically pathindependenL The variation between the different paths is of the order of 20 per cent for thedirect coarse analysis and converges to approximately 0.1 per cent for the fine grid; the variation I __
between paths is yet smaller for the regularizing procedure (25) with the differences beingactually less than 0.5 per cent for the coarse grid, 0.04 per cent for the fine. Moreover these
t See Reference 7 for a fuller dicusmion ef the advantages ef paIth iPdpedet ntelh ralveti to mch, osuh aslocal collomtion in this regard.
%~ V
*.t ..,-
PATH INDEPENDENT INTEGRALS 1007
results are the worst experienced in a number of numerical experiments since the *ins of theparticipation factors here are such that errors accrue rather than compensate. Accordingly itwould appear that the choice of integration path is not a major concern in applying the H,integral, especially when doing so in conjunction with the regularizing procedure. The numbers Z>'in Table I also indicate the sort of convergence to be expected with H, integral. To illustratethis more clearly we model the errors for each analysis with
e = eohc (26)
wahere e is the absolute percentage error in the dimensionless stress intensity factor for adimensionless grid size of h: e0 is thus the error for unit grid size and reflects the initial accuracyof an analysis, while the constant c gauges its rate of convergence. We fit (26) to the mediumand fine grid results in Table I since we expect that c is more likely to have converged to aconstant itself for these grids. To check (26) as an error model we then use it to predict theerrors for the coarse grid; with the exception of the direct calculation of H, on 11 thesepredictions agree well with the actual errors supporting the use of (26). The results of applyingit can be summarized by the average values determined. For the direct calculations,
eo -13(%M c-1-1 (27)
For the regularizing procedure results,
o- 3 (%) c-2-1 (28)
Equations (27), (28) show that for this test problem the regularizing procedure not only leadsto markedly better convergence as expected, but in addition is about a factor of four more
d accurate. Further, the regularizing procedure was observed to enjoy a similar superiority overdirect computation in other test problems and the sort of relative improved performancedemonstrated in (27), (28) is characteristic of that obtained in earlier applications of theprocedure to crack problems (refer to Reference 7 wherein average values of eo and c are 42per cent and 0-15 for a direct calculation using the J integral, in contrast to 8 per cent and1-4 for the procedure). It would thus seem that the use of the path independent integralsdeveloped here, together with a regularizing procedure as in (25), offers an attractive andefficient computational approach for singular elastic notch problems.
CONCLUDING REMARKS
The path independent integrals described furnish engineers with a useful tool for determininggeneralized stress intensity factors in elastic notch problems. This is especially so when usedwith a regularizing procedure such as that of Reference 7; then accurate results can readilybe obtained using any standard, constant-stress, finite element code. Improved usage over thatdemonstrated here can be brought about by introducing mesh refinement with the grid gradationbeing tuned to the complete problem in the instance of a direct calculation, and to the residual -
regular problem when employing the procedure in Reference 7.Extension of these integrals to treat problems involving loaded notch flanks is elementary
via superposition when such loads are constant but otherwise awkward. Extension to compositeconfigurations is possible at the expense of some algebra and is being undertaken at this time;extension to anisotropic notch problems is possible in principle but does require significantalgebra. Extension to three-dimensional geometries in which the notch tip traces out asufficiently smooth curve is also possible in the light of Aksentian's analysis" which shows thesingular character in such instances to be that of the two-dimensional problem. However, for
1. 4
1008 G. 8 SINCLAIR. M. OKAJIMA AND J. H. GRIFFIN
geometries that are more significantly three-dimensional, such as the intersection of a notchwith a free surface, the integrals given here would not be appropriate and the developmentof suitable new integrals would appear to represent a considerable analytical task.
ACKNOWLEDGEMENTS
We are most grateful to one of the reviewers of this paper for some very constructive criticismand, in particular, for drawing our attention to the work of Stern and Soni,' thereby leadingto more elegant and computationally efficient forms for the H, and H,, integrals. We alsoappreciate being given access to the code PLANDJ, originally written by J. L. Swedlow of the
4 Department of Mechanical Engineering, Carnegie-Mellon University. and the financial supportof the Center for the Joining of Materials at CMU under a grant from the National ScienceFoundation.
REFERENCES
1. J. H. A. Brahtz,'Stress distribution in a reentrant comner', Trans. A.S.M.E. UI App!. Meek), 35, 31-37 (1933)5.2. M. L. Williams, 'Stress sinsulanities resulting from various boundary conditions in angular comners of plates in
extension'. J. App. Mech. 19, 526-528 (1952).3. J. D. Eshelby, 'The force on an elastic singularity'. Phil Trans Royal Soc. (Lond.). A244. 87-112 (195 1).4. 3. L. Sanders. 'On the Griffith-Irwin fracture theory'.). App. Meek 27, 352-353 (1960).5. L. B. Freund. 'Stress intensity factor calculations based on a conservation integral', In J Solids SomaC 14.241-250
- (1978).6. M. Stern and M. L. Soni, 'On the computation of stress intensities at fixed-free comners', IAL J. Solids Smnier. 12,
331-337 (1976).7. G. B. Sinclair and D. Mullan, 'A simple yet accurate finite element procedure for computing stress intensity
-' factors', Int . ussame,. msethods eqg. 13, 1587-1600 (1982).3. R. D. Gregory, 'Green's functions. i-linear forms. and completeness of the uigenfuncuions for the elasostatic
strip and wedge'.). Elasticity, 9. 283-309 (1979).9. 1. S. Sokolnikoff,. Mathematical Theory of Elnsrcity, 2nd edn. McGraw-Hill, New York, 1956.
10. G. Strang and G. F. Fix. An Analysis of thme Finite Element Method. Prentice-Hall. New Jersey, 1973.11. 0. K. Aksentian, 'Singularities of the stress-strain state of a plate in the neighborhood of ans edge'.,). Appl. Mark -
Meek (PM.M.), 31, 193-202 (1967).%
-%4
,1,
International Journal of Fracture 27 (1985) R81-R5. 0 1985 Marinus Nboff Publishers DordrecbL Printed in The Netherlands
A REMARK ON THE DETERMINATION OF MODE I AND MODE 1I STRESS INTENSITY ..
FACTORS FOR SHARP RE-ENTRANT CORNERS 1. 0
G.E. SinclairDepartment of NechzanioaZ Engineering, Carnegie-Mel Zon LnivereityPittsburgh, PenneyiZvania 165213 VSAtel: (412) 678-2604 A J7
, •The independent development In Carpenter [1] and Sinclair, Okajima,* and Griffin [21 of path independent integrals for the stress intensity* factors at sharp re-entrant corners provides fracture mechanics with a
useful analytical tool. Prompted by Carpenter's recent sequel [3] to (1].in which he extends his earlier work so as to be able to individually %determine mode I and 1I factors, the purpose of this note is to point outthat such a capability Is already available in the integrals of (2). Thisattribute mtay well be obscured in [2] by the overstatement there thatmixed mode problems "need" to be split into their symmetric and anti-symmetric parts prior to applying the corresponding path Independent In-tegral. While this Is certainly one approach for distinguishing betweenmode I and mode 1I. It is by no means necessary. Simply applying the .
integral for mode I In [2] to a mixed problem yields the mode I Intensityand vice versa. This Is basically because the complementary fields In e.,the mode I integral are symetric so that when they are multiplied by any 'antisymmetric fields, Including the singular mode II field, then inte- *..,
grated, no contribution results. The converse holds true for the mode II
Integral. In detail the argument is as follows. """
Consider an elastic plate with a re-entrant corner, or sharp notch,which has stress-free flanks (Fig. 1). Let x , x be rectangular
cartesian coordinates aligned such that the ntgatve x -axis bisects the
opening angle at the corner into two angles of m- . For thisgeometry we can define the respective stress intensity factors for mode 1.mode 11 by
Lim /(2v) x a K-m (2 1)x a" 1 2-111 22()o+
on z O. In(1). its the normal compogent In the x2 -direction ofthe itress field 0 , j - 1, 2), while , are the singular algen-values steinng frM their corresponding eigenequations,
sin 2+ - sin 2m, sin 21-0 a I-sin 22 (2)
with 0 < , - < 1. The elastic fields associated with and - are,respecqively, symmetric and antisymetric about the x,-afis. That Is, Ifa.4 are the stress and displacement components fo A with 0it uI "
bling those for A',
.Tht Journ of Fracture 27 (1985)
Z&. *- "
' %
R82 .
S (x iJ a + ()i + + - - + (3)ijl 2 - Jtxl x2) ui(xx 2) -) ui(Xl, x2) 2 2.2
and
oij(X _ x2) . _ (_)i+_,j 1 2 o1 (xl" x2), ut(x1 , - 12) () I u(x 1 , 2)
r- r
(4)Now from 12) we have, as our path independent integrals for determiningthe stress intensity factors for these singular fiela6,
H1 -f (aj U I a Oj u )nds (5)
and
H -r (o u - u )n ds (6)11 ij i: i i
Here, I is any contour within the plate commencing on the lover notchflank and terminating on the upper; n are components of the unit outwardnormal to Z and do is an infinitesimal element of its length; &he 4nte-gration is to be performed in a counterclockwise sense; and o, ut arethe stresses and,4isppcements of the symetric complementary iingulareigenfunction, a , u those of the antisymetric (see Appendix fordetails). The rtdlipation factors of these last are adjusted in [2]so that they pick off the corresponding stress intensity factor, i.e.,
ft..- K. H K.(7)S1 'IHI III
Turning to their application in mixed mode problems which have bothsymmetric and antisysmetric singular fields present, we first focus onthe use of H established for any problem's elatic fields in [2], thegeneral contiur I can be exchanged for the here more convenient path t(Fig. 1). Applying H of (5) on r, then changing variables so that seg-ments in the lover half-plane (x v 0) are expressed in terms of those inthe upper, we have 2
H 1( 61_ ) _x _)u*(6, x2)' ,,.%fI - !(allXl x2)u(; 1 , x2) + oil(X1 2 1, - 2))0
-(Oil(1, x2)u,(x1 , x2) + Oil(Xl, -z2)ut(;. , 2
*Although the loer case subscripts only range over the Integers (1, 2) the
usual summAtion .convention for repeated subscripts still applies over thisreduced range.
Tnt Jow'n of fraoture 27 (1985)
". . ..-P .1,. . . . . .. . ..". . ..- .. . .- '. "." , . . . . . .. .. ' . '. ',. ' . . - ". .." '% % - -- - .J . - " . . ' . '
R83
+ (a 1 xl' X 2o)U (x 2) - ai2(x. - 2)u I (x - x2)),
x 1. *.
- (1 2 x 1 , 2 ) u ( x l ' a;2 ) - t ( x - x ' -'X' 1 2.d.
Now introducing into (8) the antisymmetry relations satisfied by the anti-symmetric parts of a , u, namely the equivalent of (4), in conjunctionwith the symmetry reiltiofil sa&isfied by the symmetric complementarysingular eigenfunctions a , u ., viz., the equivalent of (3), we see thatall the combinations in pPentieses () cancel. It follows that the onlynontrivial contributions to H must come from the symmetric parts of o, ...u . Further, as proven in [21. only the singuiar part of these symnetic,fields actually contributes, recovering the first of (7). Analogously,H11 may be shown to pick off only KI in mixed mode problems.
Thus for general problems containing a mixture of symmetric and anti-symmetric fields, in addition to the strategy of separating the givenproblem into its symmetric and antisymmetric constitVents if it occupiesa region geometrically symetrical about the x1-axis , one can simply ...apply the H and H integrals directly on the path of choice to discernK% and K.i Iespectively. Either technique provides reliable estimates oftie stre§ intensity factors when implemented numerically, and either canbe made quite computationally efficient when used together with a super-position procedure or regularizing approach such as 14].
REFERENCES .. ,[1] W.C. Carpenter, International Journal of Fracture 24 (1984) 45-58.
[2] G.B. Sinclair, H. Okajima, and J.H. Griffin, InternatinaZ Journal foroericaZ Methods in tnginering 20 (1984) 999-1008.
1 3] W.C. Carpenter, International Jour'nal of Fracture 26 (1984) 201-214.
[4] G.B. Sinclair and D. Mullan, International Journal for AerioaZ . p
Methods in Engineering 18 (1982) 1587-1600. %i* 21 Jauar2'I98S
APPENDIX
Here for completeness we furnish details of the complenentary singu-lar eigenfunctions required in the definitions of H and H in (5), (6).These fields are most readily expressed in cylindrilal poll 1 coordinatesr, 9 (Fig. 1). For the I integral we then-have
ar - r l(A + 3)cos( +) - cos(A - 1)0]
0% K*r- -[(A - l)cos(X + l)e - 0 cos(C - 1)0]
a -K r- [(A + )sin(A + 1)9 - 0 sin(X - 1)@]re
*Such a strategy offers computational savings with finite element analysis
of about a factor of four in terms of operational counts and around a fac-tor of two in storage requirements. See [2], Eqn. (10) at seq. for specificsof the decomposition.
Z.nt Journ of Fracture 27 (1985)
.-.
:. ..° .. ... .... ... .. ... ..... .... .. . .. .. .. ...- ... .... .. .. ... .. .. ... -. .. . . .. ... ..... • ..... ... .., ... ..- ..
• % 4k
I84
U . [(X + .)cos(1 + 1)e - B cos(X - 1)e]* K~r_ * -.. ."
K ru K [(A - K)sin(X + 1)0 - B sin( - 1)6]
K * .(8)UjPI sin 2 a + sin I )(1 + K)(sin 2% + 2a cos 2a)
in vhich
21
- 1, " 1 coo 2a + cos 2) o
and wherein z is the shear modulus, and K - 3 - 4v for plane strain and(3 - v)1 + v) for plane stress, v being Poisson's ratio. For theintegral we have
a -K r0[B sin(A - 1)0 - (X + 3)sin() + 1)e]
a K r 1 to sin(A - 1)6 -(A - 1)sin(A + 1)6]ee- _ _ ..-1.
o K r - B cos(1 - 1)0 -(1 + 1)cos(X + 1)0]
Ur M,(2B-A [o sin(A - 1)8 -(X + K)sinl) + 1)01
* /**r ), 1(;us. to) [ oS( - 1)e -1)k - K)CoD(), + 1)8],'O -
K I1 + 0)(sin 2a - 2* €o- 2Xa)
In which now121
AA ~ coos2*- cos 21
Like fields for the made III Integral 9 (-NIX) can be found in (21. _
6
'- .fm 2-
lrnt Jouaw of fraottw 27 (1985) 7% .I- '. r ;
* -. -. . - - -4 -. ~- ~...... .4....
U,
U, -4.
4 185
*I 4 II,
.4..-
4,. *4
I. I -
4.
I
~'NL
a
I~4. 4.
* C
*44.
j4
a
.4 4.
.4-..
p4,. -.4
-4
4,
I
.4,
'4...
~ .. ~ ~Figure 1. Geometry, coordinates, and integration paths
4.
h* 14.
mt Joww of Fraotww 27 (1985)
4.1 4
~ *1 4.I44.4
'I
International Journal of Fracture 29 (1985) R17
0 1985 Marinus Nijhoff Publishers, Dordrecht. Printed in The Netherlands.
CORRI GENDUM1-0
Correction: "A Remark on the Determination of Mode I and Mode 11 StressIntensity Factors for Sharp Re-Entrant Corners," G.E. Sinclair, Inter-nationaZ J7ozonal of Fracture 27 (1985) R81-R85.
Equations (5) and (6) on pg R82 should read:
B ./ *i a~ * U u)n ds (5)
H I I (0~ a:* *ul) n ds (6)
The paragraph following Eqn. (7) on pg R82 should read:Turning to their application in mixed mode problems which have both sym-metric and antisymmetric singular fields present, we first focus on theuse of H to determine K in such cases. In view of the path indepen-dence Of established ior any problem's elastic fields In [21, thegeneral coAtour I can be exchanged for the here more convenient path r(Fig. 1) ....
Equation (8) Is split and occurs on the bottom of pg R82 and the top ofpg R83. The arguments for the last displacement field occurring on pgR83 should read:
u I(x 1 -X 2)
We regret any Inconvenience crested for our readers by these typographical ..
errors - Ed.
12 .7uly 1985
.tnt J~ourn of Fracture 29 (1985)
b - --- -7
APPEDIX3: OMEINHEENTY URELABLEPRATICS I PREENTDAYFRATUR
MECHANICS ~ ~ ~ ~ ~ ~ ~ ~~~~- FUTE XMLSO TEURLAIIYOFSRS NEST
ESTIATESFOUN VI LOCL FITIN
-'e t7.,
International Journal of Fracture 28 (1985) 3-16.C 1985 Maninus Nijhoff Publishers. Dordrecht. Printed in The Netherlands.
Some inherently unreliable practices in present day fracturemechanics
G.B. SINCLAIRDepartment of Mechanical Englneering Carnegie- Aelion t nwer'sui, Putisburgh , 'A 12J3, L'.SA"
(Received July 30. 1984. in rised form December 7. 1984)
Abstract
* A number of current practices in fracture mechanics which use quantities near a crack tip to make conclusionsabout response at the crack tip itself are examined. Specificall) these include: stress and displacement matchingto estimate stress mntensit) factors, monitoring local stress and strain values to predict fracture, and both crackopening angle and crack opening displacement as fracture criteria B) means of a pair of counter applications, all
% of these procedures are demonstrated to have the potential of leading to completely incorrect conclusions. Anunderstanding of what causes this inadequate performance then indicates that such procedures may be unreliablein general and prompts suggestions as to alternatives.
1. IntroductionIn fracture mechanics today there are a number of procedures which in essence draw on
field quantities in the vicinity of a crack tip to infer what is happening right at the crack *
tip. This paper considers two classes of such procedures: local fitting methods for .,determining stress intensity factors, and local fracture criteria for predicting fracture. The P
intent is to examine the reliability of these approaches.Local fitting methods for calculating stress intensity factors at cracks arose out of a
need to extract this parameter from numerical analyses, particularly finite elementanalyses. One of the first discussions as to how best to undertake such exercises is that ofChan et a). [I], though certainly the approach was in use prior to [1] even if somewhatinformally. Moreover the approach, together with spin offs such as the nodal forcetechnique of Raju and Newman 121, continues to be used today in conjunction with bothfinite element and numerical integral equations analyses, as is evident in several papers inrecent conferences 13,4]. All of these methods entail matching near but not at the crack tipto estimate the stress intensity factor there; the question cons;'ered here is how reliablycan the attendant extrapolations be carried out.
Local fracture criteria attempt to predict fracture by checking some quantity in thevicinity of the crack tip. Among the more natural choices of quantity to this end aremeasures reflecting stresses and strains at the crack tip. Stresses are usually used in elasticanalyses and typically in complex configurations (e.g. for failure in composites as inChamis [5], and for biomedical applications as in Valliappan et al. [6J). " Strains are
'p.
There do exist though, vamples of the diret use of local sresses in elastoplastic treatments. sta for instanceMauer e it al. -.
.
i7*'1
-i- --.--. : . 9 f*
. . . . . . . . . . . . . . . . . ... .. . . . ..-.... ..-..... . - +. -. , . .. -. . .- +. . .: . . -.-..- -4. .... .. . . .. . .. , . .-. +..... ._ . .. .. . ... ... _.. . ....-. _ . ,.: _ _..,.._ _ ,- .-_, . . ,. , ,: . .. ... .-.--- < .. . .. ". - ' .. .c
I'
4G.B. Sinclair
normally preferred when significant plastic flow accompanies fracture (e.g Newman [8].Belie and Reddy [9]. Kim and Hsu [10]). Other quantities employed in this role are thecrack opening displacement of Wells []I] and the crack opening angle of Andersson [12].the former having gained sufficient acceptance to merit a British Standard [13] to govern .%its measurement. * All of these criteria involve comparisons made away from the crack tip% ith a 'ie% to gauging what is happening there, the issue of concern here is the certainty%ith which such comparisons are connected to fracture at the crack tip. ,_,-
We begin our assessment in Section 2 by specifying the local fitting methods and localfracture criteria for testing/calibrating and thereafter using on simple elastic problems inSection 3. The conceptual applications in Section 3 in fact have known solutions, thereb"furnishing demonstrations of local procedure performance. These sample evaluations inturn motivate a discussion of the more general use of local procedures and somesuggestions as to alternatives in Section 4. The paper then ends with some concludingremarks.
2. Procedures examined
Here we first define a stress intensity factor and describe two local fitting proceduresaimed at estimating it - one using stress, the other displacement. We then specify somelocal fracture criteria which are based on stress. strain, crack opening angle, and crackopening displacement.
To fix ideas consider a cracked elastic plate which is thick in the out-of-plane directionso that a state of plane strain obtains (Fig. 1). For the in-plane directions, we let (x. v) berectangular cartesian coordinates having origin 0 at the crack tip and x-axis aligned withthe crack. To further assist the development, we also let (r, e) be cylindrical polarcoordinates, sharing the same origin, and related to the rectangular coordinates by
x-rcos9. y-rsin . (1)
for 0 4 r < 2c. - w < < ir. Next we assume the plate to be under symmetric loadinghaving resultant force per unit thickness. P, acting transverse to the crack. Then the onlystress intensity factor present is the mode I factor K t, defined by
K",- lim 2V'rr'cU(o#) on 0-0. (2)
where a# is the elastic normal stress in the $-direction and U is the unit step function. heretaken as being one for positive arguments and zero otherwise.
We now wish to draw on local quantities to form estimates of the stress intensity factor.K,. One straightforward stress estimate is suggested by definition (2) itself and merely sets
U- on 9-0, (3)
wherein r0 is some "small" distance from the crack tip and the bar atop a, distinguishes itas being whatever value is found by the numerical solution technique adopted. Ananalogous displacement estimate is
K-Il,., V-)u( ',.,,) on 0-r. (4)
Here. is the shear modulus. , Poisson's ratio, and -ri on 9 - v the numerically-de-
Crack opening displacement was also independentl% introduced by Cottrell 1141 to the effect a somewhatdifferent objective. that of classifying brittle versus ductile fracture - ee Burdekin 115] for a mcet r.0eA of itsrole in fracture mechanics.
7• .
- . . - -.
-'- ...
Some inherentlh unreliable praclices 4. .4
A P0 o., P
Ctc.k C"'ra'
00 x -- Symrnet ry Axis I ..
Elastic Plate
I
JJ d
Figure 1. Mode I crack configuration and coordinates.
termined opening displacement of the upper crack flank. Again in (4). for simplicity, weselect r - re as our -small" distance, a policy we continue throughout the remainder ofthis discussion without loss of generality. The estimates in (3). (4) both fit the first singulareigenfunction for a symmetrically excited crack to the entire crack fields at a stationsupposedly close enough to ensure that these singular fields dominate. Thus while (3). (4)do not represent the most sophisticated of local fitting procedures for finding K1. they docontain the essence of the rationalization for all such approaches. Consequently they areadequate for demonstration purposes here; subsequently we review other. more complex.local fitting methods.
We now turn to fracture criteria based on local quantities. For a stress criterion wechoose perhaps the most obvious, namely that the tensile stress ahead of the crack. nearbut not at the crack tip, attain a critical value for fracture. That is, " .'
on 0-0. (5)
for fracture, wherein a'(> 0) is the critical stress determined on a suitable calibration l-- lwproblem. An analogous strain criterion exists. Instead, with a view to examining a greatervariety of local fracture criteria, we consider the more involved condition of Newman 18].In Newman 18]. the crack-tip strain measure is assembled from a finite-element analysis. ',-..-The result, denoted 1, here, is the average of the element strains in the y-direction. c,. ,taken over all the elements connected to the node at the crack tip. Accordingly for fracture
%W:.;,,..~ a . ~ - .. 'j -.p~~jw ~ '4'4 ..&".'4 "
7 - ..-. . . .. -
6 G.B. Sinclair
we have (Fig. 2)
3 -", 1Y (6)
where o (> 0) is a calibrated critical strain and r, - r2 F5 r./3. r3 Vr 0/3, 01 cot '2.02 -tan - '2. , -3r/4. The sample finite element grid of Fig. 2 also suffices for
computing the crack opening angle. a. of Andersson (12]. this being the angle subtendedby the closest node to the crack tip. The associated fracture criterion has that, at fracture.
a-a,. a-2tan-'( -ue!,.,,/ro) on O-r, (7)
with a,(> 0) being a calibrated critical angle. A similar evaluation of the crack openingdisplacement. 8. of Wells [11), leads to the fracture criterion that has. at fracture.
8-8. 8- -2uo,,o on 9-7, (8)
8( > 0) being a calibrated critical displacement. Actually 8 is usually evaluated as thecrack opening displacement at r,= r w, 2 /2iro?, where ,.. is the uniaxial yield stress:here we take o,. to be such that this station too coincides with r- ro . Equation (8) ...
completes our specification of the set of fracture procedures based on local quantities tobe appraised next.
\ / Finite Elemaet Grid\
_ ~'. .
-- Deforwed Crock Profile L
Figuft 2. Finite teme rd and othei fatues ear the ack tip.
3. Test/calbrtin prblem: applicatons
In this section we start by formulating a test/calibration problem and a pair of problemsto serve as "applications". Then we exhibit the closed form solutions to all three problems.employ the first of these to test the local fitting methods and calibrate the local fracturecriteria, and thereafter use the last of the solutions to assess performance.
The problems to be treated are drawn from the class of symmetric, plane strain, crackproblems outlined in the previous section. We do this conveniently by confining attention , -..
to the cracked circular plate of radius A (Fig. 3) and hence, by virtue of symmetry, need I -only to consider the semicircular region 9, defined by
D,, ((r. )1O <rc R.O <90< r). (9)
Throughout this region we seek, in general, the stresses e,. as. -, and displacements u,, u"as suitably smooth functions of r, 0 meeting the following requirements. All threeproblems are to satisfy: the plane-strain stress equations of equilibrium in the absence of
- a -. -S-. _
i~~q
,- ".-;
r~~u M T 7'
*%*
- Some inherenh', unreliable practices 7 ".
5"0 -.. ":
\
' F
Figure 3. Cracked plate geometry for test problem and applications.
body forces. namelyMI., + TW +0,,- 0, -0 (.10
a#.# + rT,,., + 2,, - 0.
on ,. where subscripts preceeded by a comma denote differentiation with respect to thatvariable: the plane-strain stress-displacement relations for a linear elastic, homogeneous eand isotropic, plate with shear modulus p and Poisson's ratio v - 1/4. vi.."
Or M[3",, + r-'(,,+ ue0,1] i:a# =,[3r-'(u, + u#.#) + ,,, ,].()"''"
on 9.; the stress-free crack flank conditions which seta# -*r,# - 0 on 9-i, (12)
for 0 < r R; and the symmetry conditions ahead of the crack which haveNo -O, r,-O on 0-0. (13)
- for 0 < r < R. In addition each problem individually must comply with its applied stressconditions on the circular boundary. These set %7.
Ppr-. ,- " an cosT on r-R,
" , n-O.a 0.... .ms ~(14)'.-"P no
T b sn--- on r- R. (14)
for 0 < 0 < w(P > 0), where, for the test problem.
k-, a-S, a3. -1, bl-b 3 -1 (15)
are the only a. -0 0, b. -0 0. and. for the two applications, all nontrivial constants are given", in Table 1, wherein p -A/r 0 (> ).
While not strictly needed in the mathematical formulation of the foregoing type ofelasticity problem. we adjoin the following regularity requirements on the displacements in
% %%%
-o o
... ./. ." - " -. .-- .".-.-.. -. - ':'2""':,: ,- - - ' :" ---"9 .- . .? --.-, '.-.i --,--- - '. -' - . '/ ,: ""
• " 8 G.B. Sinclair , P,
Table 1. BoundAr. condition constants for the applications
Constant Application I Application 2
8a, (p - 2 + v2 )2/63p2 B( o - 1)2
o,. a, - 7.374.036p /2 - 5.969.792pi 2 % .a 0 N " -
a,. h, 0 156 0.9l8p'1, 10(9- 4 / )p'/49 - I - lOp,
a, 2(100+ 4 )p/7 0h.-A. 0 47.012.754p'a. 3(a3 - all/'7) - 3P2(10 +.,9p:2 ./_
,. - 0 10.008.616p5 2, u. ,.A -A , -So. 0 .-
; - all ,. -sil 9( 11 , la " 67;": ..
-1 0 1b., 3a, 1 - 30p.
, -3a'a, 3p 2 + a,12
all three problems to emphasize their physical admissibility: that the displacements at thecrack tip be bounded, that is
u,-O(1). u.-O(1) as r--o (16)
on -4. where 0 is the large order symbol: and that the crack opening displacement benowhere negative. i.e..
-u,; O on - r. (17)
for 0 < r < R.We postpone for the moment a discussion of how the test problem and two applica-
tions were arrived at together with their solutions and merel) exhibit the latter at thispoint. For all three problems the complete solutions admit to representation by finite .". ,,sums of eigenfunctions for the crack, viz.,
a,-- c.rx'- b / 2 (6- n) cos(n-2)! +(n+l-)2)cos(n+2)kv - ... 2 ''
o=- CR r." n+2)1cos( n- 2) -1 + -"2)€os(1,,+2) .
-';7V 2, 2
VrR oi ..... "" "
P 17
k.- . 2.,
P9
Al-
x f.Crt"/-'1:(4-n) cos(. -2) + (n+l-)'2) cos(n +2)!""--.2.... IL %
SX E --r( (n + 4) sinln - 2)! -(n +1- 12) sin(n + 2)!i¢. 2- . ...i ~ ~on R. For each problem respectively. the solutions have k as previously ((15). Table 1) :
A- %
..
,7 .- '-4-
-• X" V-
Some inherenilv unreliable practices 9
Table 2. Coefficients in the solutions for the applications
Coefficient Application I Application 2
ci 0 l2c, -1.943.509 -1.492.448 ", -
'412 0 7.845.459r109 - 4 V12)/49 -10S,, -. 0 1.251.077
*'t", -2(10+ 2 )/7 0
a, ,, 9011. - 6v )/49 9
while the only nontrivial coefficients in the above are: for the test problem,
c- 1; (19)
and for the two applications, those c, ,6 0 in Table 2.The solutions in (18). (19). Table 2 may be verified directly. Substituting the forms in
(18) into (10). (11) establishes that the field equations are complied with, and inspection of(18) shows that the crack flank and symmetry conditions (12), (13) are met. Evaluating q,, - -
.r, of (18) on r - R. using (19), Table 2, further shows that the boundary conditions (14),(15). Table I are fulfilled. Finally, checking the displacements in (18) under (19) or Table -2 reveals that the regularity requirements (16), (17) are satisfied.
In applying the local fracture procedures to the three problems we begin by assumingthat their numerical treatments have been refined to the extent that the exact answers arerecovered to all intensive purposes. i.e. # - ot, etc. This is the ultimate of situations froman analysis point of view and allows us to focus on whatever errors are introduced by theprocedures alone.
Invoking this assumption we next test our local fitting methods before applying them, a . 4
precaution commonly undertaken in practice. Using the local fits for the stress intensityfactor in (3), (4) with P - 1/4 on (18) when (19) holds yields, for both methods,
4: P 7(20)
For comparison, taking the defining limit of (2) in (18), (19) gives
K, =4 P L~w(21)
Hence the local fitting values of K, are exact and the performance of the two methods (3),(4) on the test problem is perfect.
In order to calibrate our local fracture criteria we need to determine the load to fracturein our standardizing problem. We do this by regarding fracture as occurring when thestress intensity factor takes on a limiting value of K , the fracture toughness for the plate.It follows from (21) that the critical load in the calibration problem, Pr, is
74 "(22)
Thus increasing P to P, and evaluting the stress in (5), the strain measure of (6)'. the
For this derivation we ote that trailhtforward manipulation leads to r-
-"' .-. 1(2- n-(-)'2)cos(e -2)! +(-2) cos(n--6)
a as the normal strain in the ydirtction to be calculated at the set of points in (6) et seq.. and threafter avera gedto produce d,.
%
•." . . . - . ""-"9. """" ', . . . . . . , , ,""" .-. ,,,.." .'.',, . ,",. .+.'.".: '. .+.".•.','.".'.-.'.-.-, ' .,.+,-," ." '.
10 G.B. Sinclair .-
crack opening angle (COA) of (7). and the crack opening displacement (COD) of (8).using (18). (19) gives, respectively. V
4P, Xe ,
4P~k' K~k'
kp roR PM2 'rro(23)
6P 3K,3a- 2 tan - __ -2tan - __ -'a,
pk~roR 2gtj2rr 0
2 2Pe. 3 K""#L" ' R+ r:T:'::- a
wherein , has been set to 1/4 and k'- 0.303,299. -'.With our local fitting methods tested and local fracture criterion calibrated we can
apply them to the two applications. Introducing (18), together with the coefficients ofTable 2, into (3). (4). (5), (6). (7) and (8) with P - 1/4 and when P - P,. of (22). with ktherein now taking on the respective values in Table 1, furnishes the results displayed inTable 3. As opposed to normal engineering practice, we can also determine the corre-sponding exact answers for the stress intensity factors in our "applications" from (2). (18)., .
and Table 2. and thereby check the values of K, when P - P. These last results are alsoincluded in Table 3.
Table 3. Comparison of procedure outcomes with actual results
Acuiity Application I Apphcation 2
K, determiuation
Stress fit K, 0
Sam fit
Exam answer K,0" ..-.0 :
fracture prediction
Local stain 1, - cc 1,-0COA 8, 0-0COD 8-8, 1-0Exact anwer P - P, .k -0 P - P, K, -K,
Several comments on the results in Table 3 are in order. For the first application, thestress intensity factor in reality is zero irrespective of load level P so that the stress right atthe crack tip is also always zero and fracture there is not a possibility (at least not prior tofracture elsewhere): the local fitting procedures on the other hand estimate Kj as being "dependent on P and consequently, when this load is sufficiently large (P - Pc in fact). all Vthe local fracture criteria predict fracture at the crack tip. Hence the procedures consid-ered give rise to conservative but gross errors in Application 1. For the second applcation. "4. - . "
the stress intensity factor is not zero and increases linearly with load so that there isindeed a load level at which fracture occurs: the local fitting procedures in contrast giveno K, whatover, independent of the value of P, and companion predictions of nofracture for any load. Hence the procedures give rise to nonconservative gross errors inApplication 2. In all the results in Table 3 represent a clear demonstration of the potential '.%. .
I.v'.
;. :. +, .. ..;' +. .. / .- +. .-. -. .,- ., > -. ..-. .- .. . .. ,- - - .-'-. .. ... + -. .. .. ., .. . " ,. .& ".. +, .+ = . .. ..-. ... ... .
Somre inherenty unreliable practices 1
unrealiability of the fracture procedures of Section 2 - we look for means of controllingsuch unreliability next.
..:.? +
4.~~' Som°.teratve
Here we start by examining the nature of the breakdown of the two local fitting methodsused to estimate K1. In the light of this appraisal we reiew other local fitting methods andadvance suggestions for improved methods of extracting stress intensity factors. Finallywe discuss what remedial action can be taken for the local fracture criteria.
The reasons underlying the failure of the K, estimation via local fitting in the precedingsection are as follows. First, the fit must be made away from the crack tip with r0 , 0.since the stresses can be singular and thus not fittable at r0 - 0 w+hile the displacements p.
are zero there leaving nothing to fit. Second and as a consequence, other eigenfunctionsfor the crack that are not being fitted can participate causing incorrect answers. Indeedthis is the effect that the applications in Section 3 were constructed to produce. That is, -symmetric eigenfunctions for the crack other than the singular one being fitted were addedso as to generate the erroneous K, estimates of Table 3, then whatever stresses thecombination so devised realized on the circular boundary were taken as the prescribedstresses there.
At first glance it might appear a simple matter to design a local fitting method which ', -does not succomb to such contrivances. Perhaps the simplest candidate stategy to this end t -4is to move the point fitted closer to the crack tip; such an approach however fails if ro inthe counter applications is then adjusted to coincide with the new point (which it can be).A more sophisticated possibility is the procedure in which v2vra* on 9 - 0 is plotted as afunction of r, a straight line fitted to the data, then the line extrapolated to r - 0 toestimate K1. This type of extrapolation technique in effect matches the first and thirdeigenfunctions for the crack and would appear to enjoy some popularity in present day j .practice. There are a number of ways in which the details of this and like extrapolationmethods may be implemented with varying results, but typically such an approach doesmeet with more success on the given applications than that reported in Table 3. In fact. onknowing the answers, it is possible to adjust the technique to recover the correct resultsexactly. For such a fine tuned method though, it is then possible to devise a further newapplication on which it does not work by superimposing even more eigenfunctions. Byway of illustration of this sort of more extensive counter ;pplication, consider Fig. 4. "" InFig. 4, v2w- o,/ on 0 - 0 is given as a function of r/R on a set of 17 points; here K is anondimensionalizing, but otherwise arbitrary, constant. A least squares, straight linethrough these data points passes through the origin, yielding A1" -0. In actuality. K - K.illustrating the unrealiability of this approach. Further, gathering more data on 1/9 C r/R .-
4 1 and matching does not improve the incorrect estimate of K,. Furthermore. similarlyerroneous results can occur when using the displacement counterpart - indeed, there is a Nel
counter application which has exactly the same data as that depicted in Fig. 4 but with"' "ro#/K on 9- 0" there exchanged for "-w/2r pu#/K(1- ,) on - ", thus .'-
producing A, - 0 despite the fact that K, - K. tAnd this is the situation that prevails ingeneral. That is, irrespective of how apparently refined a local fitting method is and whatit fits, once its specifics have been decided on it is then always possible to generate anexample on which it proves to be comprehensively inadequate. The reason for this is that
SIn this poc the t bo of a - 1/4 i limply mnde to facilitate the emstncuo and not of any uaignficancem itacl f ,. . - ,
S Sincir [16 1 for d&Is of is exule.SaM 116) for deuls of that wesondrumple and othe. V'.
:::
.. *".,.., .
12 G.. Sinclair
*" .A'. -
9
7- 0
S. 8,. -• ...-v.
.. ~ b 4-•"?2 0
010
3 0 '.'
'p 00 i i iq% .
0 I/9 2/9 3/9 4/9 5/9 6/9 7/9 8/9 I
r/RFigure 4. Product of sru ahead of the crack times the square-root of distance from the crack in a moreextesve Counta applcation.
any local fitting procedure must match a finite number of quantities on a finite set ofpoints and thereby can only fit a finite number of eigenfunctions; thus any local fittingprocedure leaves an infinity of eigenfunctions unfitted whose participation factors can beadjusted to ensure its downfall.
A key question that arises at this point therefore is who has the final say. the analystmaking adjustments to his local fitting method so that it does produce good results or therascal generating the problems on which the given method does not work. Unfortunatelyin practice the latter always has the option of going last. This is because in practice we .must ultimately stop checking our procedure on test problems and apply it. Then. sinceapplications are problems for which we do not know the answers, we cannot tell with ,,certainty how well our procedure really works, and as a result we have to tolerate thepossibility of it giving erroneous estimates, maybe even grossly erroneous estimates.
Logically one only needs the possibility of such results in an application to establishthat a method is unreiabl,.. In practice, however, one can perhaps rationalize still usingthe method if one can Yssign a sufficiently low probability to the occurrence of thedisastrous example - the counter application being pathological to a degree. Certainly the %
....... ....... ......
_ X 7
Some inherent& unreliable practices 13 r .p ...
counter applications provided in Section 3 do not appeal as physical problems. beingconstructed more towards dramatically underscoring how much in error the selected localfitting methods and fracture criteria could be. Even so, the two particular counterapplications are one of an infinity of pairs that could be devised to ensure that the -methods in (3). (4) give the same completely incorrect estimates as in Table 3. Moreoverthere are other such sets of pairs that give rise to estimates that are more subtly in error .". Pthan those of Table 3 but are nonetheless significantly inaccurate And such is the case forany local fitting method, no matter how apparently refined - one can alwa\1 construct anindefinitely large number of applications on which it fails. It therefore appears unlikel\that the probability of encountering a problem application in engineering can safel\ betaken as being negligible.
More specifically in this regard, consider the following limited simulation of perfor-mance for the simple stress-fitting method of (3) on two problems that could reasonablybe viewed as being of practical consequence. In accordance with normal practice, we beginour simulation by testing our method. As a test problem we take the classical Gnffithgeometry of a center-cracked infinite plate and consider the case when the crack is openedby a uniform far-field tension. Then we adjust r0 so as to ensure acceptable results whichhere we take as being up to 5% in error. Picking to - a/14. a being the semi-crack length.as a moderately close point at which to obtain accurate stress values by either numericalor experimental methods, we find on assuming negligible errors in the stress determinationthat
, "1.05 K1, (24)
wherein K, is the exact value. Equation (24) represents a satisfactory and encouragingresult. Now as an "application" we take the Griffith crack opened by a uniform pressure.Applying (3) with ro - a/14 as before then yields
n0.67K,. (25)
Equation (25) represents an unsatisfactory nonconservative result, showing that theprobability of meeting a problem application in practice for this simple method is notzero. .
Of course, as remarked earlier, it is possible to adapt ones local fitting method toovercome the difficulty experienced in this last elementary example. However, whileincreasing ones efforts considerably when applying local fitting procedures by performingthe matching in a variety of ways at a number of locations does tend to increase thechances of detecting spurious results, it does not appear reasonable at this time to assume ....-.that any such combination of techniques reduces the probability of unacceptably incorrectresults to a negligible level in all situations of practical interest. Continuing to use localfitting methods to determine stress intensity factors in engineering would thus seem to beunjustified and unwise. This is especially so since there are available quite differentapproaches which are free from the potential of furnishing useless results that local fittingmethods have, and which can be implemented with no more effort.
One of the best of such alternative approaches employs path independent integrals,since the integral operators involved are essentially orthogonal to all of the eigenfunctions Ii.for the crack except the singular one. That is, contributions to these integral; from thenonsingular eigenfunctions can be shown to be zero analytically **. so that these terms are
See Efus et W. 1171 for further discussion of the effects of the presm of stresses which are independent of P,lke the additional hydrostatic presaure here. See also Sacilair and Mullin (191 for a further eample. concerningthe standard confiuruon of a msigle edge motch under tension. wherein ocAl fiting iethods have beenresponsible for less than satisfactory nsults.
See e.&. 1191 pp 1002. 1003 for a proof.
." ..- ,, W
"'S.--.-~--.- .-.-
;. t. . ,
14 G.B. Sinclair
in essence eliminated rather than merely hoped-to-be-small as they are to varying degreesin local fitting methods. Two such integrals are the J integral of Eshelby [20]. Rice [21]and Sanders' integral [221; both of these enjoy physical interpretation as the energy releaserate at the crack tip in the elastic instance. Other such integrals are those of Carpenter 123]and of Sinclair et al. [19] these last lack the appeal of a direct physical interpretation butcan be expected to perform somewhat better numerically since they involve the calculation ,gs, _of single terms rather than products. * In addition there are also methods, like thatdeveloped by Parks [24]. which effectively employ a path independent integral and
consequently realize similar performance.Turning to fracture criteria based on local stresses and strains, we find the situation is
not improved in terms of their likelihood of furnishing correct information. Again one canconceive of more complex criteria than the simple stress criterion and the average straincriterion specifically examined here; for example, the condition of Belie and Reddy [9]which checks the maximum strain in all the elements near the crack tip to decide iffracture occurs there. Again too one can construct counter applications on which suchapproaches fail; for example, Application 2 actually has ,- 0 at each centroid of theelements sharing the crack-tip node (Fig. 2), so that Belie and Reddy's criterion applied tothese elements would predict no fracture despite the fact that crack-tip fracture occurs inthis application when the load is sufficiently large. Simply put, these methods cannot %reliably infer what the physical stresses and strains are near the crack tip given anunderlying theory which can lead to infinite stresses and strains at the crack tip inresponse to infinitesimal loads. Good alternatives here await the development of stressand strain fields for cracks whose physical relevance is unquestionable. In the meantime,the mathematically singular stress and strain fields, while not a physical reality, are areality of analysis which must be faced. even in the elasto-plastic instance. 00 At this pointthe accepted way of doing this, at least in the elastic situation, is by means of theassociated energy release rate, whence the stress intensity factor. Accordingly localstress/strain fracture criteria must be interpreted as trying to estimate stress intensityfactors, whether wittingly or otherwise. It follows that such practices are every bit assubject to the uncertainties of explicit estimation via local fitting methods discussedpreviously, and we recommend that alternatives, such as those suggested earlier, be used intheir stead.
The last two local fracture criteria could both b4 regarded as using measures of theloaded crack profile and thereby avoiding the singular stresses and strains to some extent.Nonetheless their performance in the elastic regime is impaired by the same shortcomingthat all of the preceding local procedures have, that of not being able to reliably assignwhat proportion of their totals are due to benign nonsingular eigenfunctions and accord-ingly what part is not. As a result they cannot reliably estimate the accepted governingparameter, the stress intensity factor. I Furthermore, being fits of a single parameter at asingle point in effect, they cannot even be readily supplemented to reduce the chances ofextraneous results. In the light of this inadequacy of both measures in treating elasticresponse it would not appear reasonable to entertain with any great confidence their useto treat fracture when significant plastic flow is present, the end to which they wereoriginally proposed. Especially since, physically, elastic response must precede plasticirrespective of how much plastic flow ultimately accumulates. In the absence of a sure
* This has in reality been found to be the case in limited numerical experiments undertaken to date. refer 118.19]See, for example. Hutclunson 1251 for an analysis demonstrating the persistence of ingular behavior within"'K" ".
the deformation theory of plasticity.In passing we observe that. in view of the uncertainty in the elastic situation of COD's relationship to K,.
hence to J. attempts to find a truly general simple relation between A and J in elasto-plastic instances %ould ., ,seem to be futile.
%...
J . r.-
Some inherently unreliable practices 15 .e
connection to a physically reasoned explanation of fracture then, these measures remaininherently unreliable and others are to be preferred. The best of these at this time wouldseem to be the stress intensity factor itself and the J integral, both of which at least have -the attribute of being clearly related to the energy release rate in the special case of pureelastic response.
S. Concluding remarks .yt.
Fracture mechanics today continues to be faced with the task of making physicallysensible interpretations of the nonphysical singular fields which exist at crack tips, be the .p,
situation purely elastic or elasto-plastic. A prerequisite to success therefore is an accurateassessment of the degree to which these singular fields are present. The proceduresconsidered here - local fitting methods to estimate stress intensity factors and local.-fracture criteria to predict fracture - fail to do this in a way which can be shown to bereliable.
The basic reasons for the potential unreliability of the local procedures are as follows.First, all the local procedures must consider quantities near but not at the crack tip.Second. at such stations fields other than the key singular ones can contribute. Third. theextent of such participation cannot be either completely controlled or fully accounted for.As a consequence. for any given local procedure there exist problems on which it producesunacceptably erroneous results. "-"
The question then arises as to how likely is one to come upon such problem problems _in practice: the answer unfortunately is not obvious. For local procedures based on a ,, -
single parameter such as one point stress or displacement matching for K1. a critical stressat a single station, COA, and COD. the likelihood would appear to be quite high. Forlocal procedures employing a multiplicity of quantities such as some of the more extensivematching techniques to estimate stress intensity factors, the chances would seem to bereduced. However, for the best of these approaches in this regard, the probability ofinvalid results cannot be shown to be zero. Consequently in real applications wherein theanswers are not known, one cannot be sure that such procedures do in fact find them.That is not to say that local procedures may not work on occasions, and even quite well, itis just to say we cannot be certain they do. Accordingly local procedures must be viewed ." "as being unreliable at this time.
Acknowledgements
It is a pleasure to acknowledge the benefit of discussions of this work with J.H. Griffinand P.S. Steif of the Mechanical Engineering Department at Carnegie-Mellon University.The financial support of the Air Force Office of Scientific Research is also appreciated.
References
[II S.K. Chan. 1.S. Tuba and W.K Wilton, Engineering Fracture Mtechanics 2 (1970) 1-17.121 I.S. Raju and J.C. Newman Jr.. Engineenng Fracture Mechanics 11 (1979) 817-82913) Numerical Methods in Fracture Mechanics, Proceedinp of the Second International Conference (S'ansea.
Wales) Pinendge Press. U.K. (1980).14] Fracture Mechanics Fourteenth Symposium - Volume I Theon and Anahsis (Los Angeles. California)
ASTM Special Technical Publication 791. Philadelphia (1983)151 C.C. Chamis. Failure Criteria for Fdiameniar Composites. NASA Technical Note D-5367. National
Aeronautics and Space Administration. Washungton. D.C. (1969-161 S. Valliappan. S. Kjellberg and N.L. Svensson. Proceeding, of the Internarionat Conference wi Fint.e
Elements rn Biomechanics (Tucson. Aniona) 2 (1950) 527-..--
.......-. ;...-
5''..........
16 GB. Sinclair
17] R.E. Miller Jr.. B.F. Backman, H.B. Hansteen. C.M. Lewis. R.A. Samuel and S R. Varanasi. Computers andStructures 7 (0977) 315-326.
18] J.C. Newman Jr.. in Ci cie Stress.- Strain and Plastic Deformation Aspects of Fat igue Crock Gro% th, ASTMSpecial Technical Publication 637. Philadelphia (1977) 56-78.
19) R.G Belie and i.N. ReddN. Computers and Structures 11 (1980149-53.110] Y.J. Kim and T.R. Hsu. International Journal of Fracture 20 (1982) 17-32.1111 A.A. Wells. in Proceedings of the Crack Propagation Simposiuni (Cranfield. England) 1 (1961) 210-230112] H. Andersson. Journal of the Mechanics and Ph.1 sics of Solids 21 (1973) 337- 356.(131 British Standards Institution, Methods for Crack Opening Displatentent ICODj Testing. BS 5762 (1979)1141 A H Cotirell. The Iron and Steel Institute. Special Report Number 69 (1961) 281-296.115) F.M. Burdekin. Philosophical Transactions of the Rotaol Societ) (London) A299 (1981) 155-167116) G.B. Sinclair. "Further Examples of the Unreliabilit, of Stress Intensit, Estimates Found via Lical
Fitting-. Report SM 84-13. Depaniment of Mechanical Engineering. Carnegie-Mellon Lini..rsit'.. Pittsburg.~(1984).
(171 J. Eftis. N. Subramonian and H. Ltebossiiz. Engineering fraciwe Aftjwhna3 9 (19') 189-Z10(18] G.B. Sinclair and D. Mullan. International Journal for Numnerical Methods in Engineering 18 (1982)
1587-1600.119] G.B. Sinclair. M. Okajima and J.H. Griffin. International Journal for Numerical Methods in Engineering 20
(198.4) 99-1008.(201 i.D. Eshelby. Philosophical Transactions of the Rot-af Socier) (London) A244 (1951) 87-112. 2
1211 J.R. Rice. Journal of Applied Mechanics 35 (1968) 379-386.(22] I.L. Sanders Jr., Journal of Applied Mechanics 27 (1960) 352-353.(23] W.C. Carpenter. International Journal of Fracture 24 (1984) 45-58.(24] D.M. Parks. International Journal of Fracture 10 (1974) 487-502.(25] JW. Hutchinson. Journal of the Mechanics and Physics of Solids 16 (1968) 13- 31.
Resume
Ott examine diverses prtiUqueS courantes en Micanique dle Rupture qui font usage de grandeurs parametinquesau voisinage de l'extrifnute d'une fissure pour tirer des conclusions stir cc qut se passe a cette extrernite mimeCes grandeurs 6ont notatrnent: let valeurs de la contrainte et du deplacement correspondant a un facteurd'interisht de contrainie estime. r'enregIStresnent des conraintes et deplacements locaux en vue de predtre larupture. et I'angle et le deplacement d'ouverture d'une fissure en tant que criteres de rupture.
En constderant une Paste d'applications divergetes. on demonte que toutes ces procedures risquent 5d'entrainer des conclusions completement incorrectes.
Une analyse des causes de ces inadequattons montre que ces proc~dures peuvesit se reveler peu fiables en w4general, et conduit A des suggestions de proc~dures alternatives.
%\-p
% FURTHER EXAMPLES OF THEk UNRELIABILITY OF STRESS
INTENSITY ESTIMATESp FOUND VIA LOCAL FITTING
G.B. SINCLAIR
SM 84-13 May 1984
s4
r. 1
FURTHER EXAMPLES OF THEUNRELIABILITY OF STRESS INTENSITY
ESTIMATES FOUND VIA LOCAL FITTING
INTRODUCTION
The inherent unreliability of local fitting methods for determining stress intensityi %.....
factors is established in 11. 2]. In [1, 2], examples of this shortcoming are limited
to very simple fitting procedures, although the general nature of source of the
" difficulty is discussed: the intent in this brief supplementary report is to furnish the
details of some more extensive demonstrations. We begin with a precise statement
of four chosen problems, then exhibit their complete closed-form solutions*. Using
these forms we next extract the exact values of the corresponding stress intensity
factors and compare the results with those obtained by fitting stresses and
displacements using several eigenfunctions. The report closes with a brief discussion
of some of the implications of the examples regarding even more involved local
fitting procedures.
FORMULATION OF PROBLEMS
The plane region of concern in all four problems is the cracked circular disk of
radius R a 9 (Fig. ). To describe this geometry, we take rectangular cartesian
coordinates (x, y) with origin at the crack-tip and x-axis aligned with the crack face,
together with cylindrical polar coordinates (r. 8) related to (x, y) by
x-rcos 6,yursin (O r<o,- < IT), (1)Sir).
Since excitation is restricted to that which is symmetric about the x-axis, we can
confine attention to the upper half of the disk, R, where
R - {(r, 9)j 0 < r < R, 0 < 8 < T)., (2)
We seek then, the stresses or, o 8 , 'r6' and displacements ur, u 9 , as sufficiently
smooth functions of r. 9 throughout R, satisfying the following requirements. Thestress equations of equilibrium in the absence of body forces,
%. %.
#n the interests of brevity, no information as to how these particular problems were constructed is given: in
* this regard, see E 1. 2] for the basic approactK
'-.
\3.. .
2
Fi ur . C r c e pl t-eo e rpad c or i a e
3
'Da lare ____0_
8r r 8a& r -
(3)
r ae ar r
on R. The plain strain stress -displacement relations for a homogeneous and isotropic,
linear, elastic material with shear modulus p and Poission's ratio V,
2p au I a U. ur~ ~ V_2 _V r ra
2p1lue U auT-_V_(1V)(T iD T)r ar(4
au a-ue U919 ra C)0 ar r
on X. The stress-free crack-face conditions and the symmetry conditions ahead of the
crack,
U 0~u. T 0 when 6 - . (5)
for Ocr<R. The stress boundary conditions on the outer circular boundary which
* prescribe
K B ,(n-) '
a acs((2 )G)whenr
K (2n-1)8
)r bs2n( when r R, (6)
for 0<6<1r. where K is a constant (K>O) and a . b (n *0, 1 .... 8) are given in Table 1I5for the respective problems (1-4). By way of illustration. Figs.2 display sketches of
these stress distributions for Problem 1 (Figs. 2a and 2b share the same scale). And
finally the regularity requirements which insist that the displacements be integrable at
* the crack tip and that there is no overlapping of the crack flanks,
. % Le
4
A ,
,- r=
I'tlq
Figure 2. Applied stresses for Problem 1: .---A r 'ra 1K, B - T'rB I r-i K.'.
' * - >'..'
5Z
U 60). u,= 0(0) as r40, (7).
for 0:50<7T, and
U8 :5 0 when 8 7T, (8)J '.. ,
for O<r-R.
TABLE 1: BOUNDARY CONDITION CONSTANTS
Constant Problem 1 Problem 2 Problem 3 Problem 4
a0 (=b0 ) 420 420 14,700 -420
a -569 919 0 5067a 2.815 14,215 217.219 14.250a- 5.962 -43.222 1,387.200 -41,781 .
a3 -9.675 -121,275 -9.199,158 -121,275as -20.115 97,065 9.782,400 97,065a6 45.525 382.725 14,366,268 382,72517(: - b?) -19.683 216,513 -30.093,120 216,513as(- b8) 0 0 13,544.091 0
b -213 779 0 1,6898.585 42.785 651,657 42,750
b3 -30.802 216,606 -6,936,000 215,661b4 33,975 339,975 22.333,578 339,975b 11.367 -837 -23,157,120 -837b6 -42,525 -382,725 -9,441.144 -382,725
6 1
PROBLEM SOLUTIONS
"* For all four problems the complete solutions can be expressed as finite sums of
. eigenfunctions for the crack. Explicitly we have
K 6 cr n 5 1 r 3r * fr [(- n) cos(n - 8 (n - .) cos(n 9].
K c rl 3 1 1 3 VAE n " --- H( * -) cos(n - ) -(n -) cos(n + )8], "-.;8t177 c 2222
K 6 c r" 1 1 3r8 "purubn - )[ sin(n -)8 - -in " )1 . (9)
f277 :n-;:'2 -
6
___ n 4V~) cos(n + (n co~r c2i2' £n1) 2 2- 2
| I'
K 6 c 7 1 1 3
- - N n - 4) sin(n -) (n -- )sin(n -)6.c(2n 1) 2 2 2 2
where c, c, (n=0,1....6) are constants having values for the respective problems as
given in Table 2. That (9) and Table 2 constitute the solutions to the four problems
" posed earlier can be verified directly. That is. the forms in (9) for each n
individually can be shown to satisfy the governing field equations (3). (4) by
substitution and the crack-face and symmetry conditions (5) by inspection, while the
combination realized on the boundary r-R-9 using the values in Table 2 can, after
., come manipualtion. be shown to comply with the prescribed stresses there (6), Table
1. In addition, inspection of (9) shows that the displacements are bounded ar r-0, i.e.
(7) is satisfied, and combining the forms in (9) using Table 2 gives positive crack-
,' opening displacements, viz., (8) is met.
TABLE 2: SOLUTION CONSTANTS
Constant Problem 1 Problem 2 Problem 3 Problem 4
c 1.890 1,890 595,350 -1,890
c 945 945 0 0c -<1> -744 2.232 0 5,067c€ 950 4,750 651,657 4,750
-230 1.610 -462,400 1,610c 4 25 225 121,615 225c5 -1 11 -13,760 11
5,5
. c, 0 0 563 0•
S6
STRESS INTENSITY FACTORS: EXACT AND ESTIMATED VALUES
Defining the Mode I stress intensity factor, K1, by
K, • or o' (10)
on R. we have from (9).
• --
P,% " -. ,%.. - • d I .• .-. %.% I . L ". *%
7 ..
2c 0 1)K, K(- 2 ).
Thus from Table 2 we obtain the exact values of K recorded in Table 3 for each of
the problems. ~
As out local stress-f itting method we match the transverse stress, 0',9 on the line
of symmetry ahead of the crack (8-0) at the five radial stations ral, 3. 4, 7, 9. To
do this we take the five eigenfunctions for the crack given by (9) with n=O, 1, 3,2. 2,
5/2 therein". Then, in view of definition (10). we must solve the 5x5 system of
equations
K d *dr d 0~r12 d dr 2 d dr 5 /2 urff 7'0,8 . (12
r a1, 3, 5, 7, 9.
where Y)is our stress-fitting estimate of the stress intensity factor and dn (nel-4) are .%
estimated participation factors for the other eigenfuntions.
As our local displacement- fitting method we match the crack opening displacement,
-uon the crack flank {6=iT) at the five radial stations r al. 3. 5, 7, 9. To do this -we take the five eigenfunctions for the crack given by (9) with n-D, 1, 2, 3, 4*
* Then it follows from (10) and the forms in (9) for n&O that we must treat the Sx5 *'
system
2 3 4 J~ 6(i d~r d dr d dr ~d r
(13)r a 1,3, 5, 7,9,
where I(i is our displacement -f itt ing estimate of K, and dn (n a1-4) are estimated
* participation factors for the other eigenfunctions.
~The eigenfunction associated with me1/2 is not as given in (9) since a 00 on 6=77' when nal/2 in (9).The isigent unction fot this gigenvatue requires individual analysis and eventually simply reduces to a equal to aconistant as the only nonzero stress mn the piano. As a result 01 so on 89&D for this case so tJAt in effect
thi eienuntonis also being matched by our stress-fitting rc re
.e mere the actual eigent unction associated with n - 1/2 has u -0 en 13 7 and so contributes nothingto a crack opening displacement fit (just as the corresponding striiss does not affect a stress fit). Further.although (9) does give the correct eigenfurictions for m=312. 5/2. 7/2. 9/2. none of these terms add to u6 Pon 6.77' Hence in effect here we are fitting the first ton eigenfunctions.
.0iv
8
We now apply our local stress fitting method to Problem 1. From Problem 1, from(9). Table 2. we have
B, 6 2c
Sn.
248 r2 r3=K[1 - i -r -(190 - 46r 5r2 *-J (14)
35 189 5
Hen:e system (12) becomes
K d, d2 d d K,K, 3d, 34rd2 Sd3 + r 3K.
2 3 5~d (15*56d, 5J'd2 - 25di3 + 5~ 4 5K, (5
SK, +7d, 7 7 d2 +49d3 . 49d 7K,
9S*d d 27d 2 *81d 3 + 243d 4 9K
Solving (15) yields K .m d 1 a K. d2 *d d4 30; the first of these is the estimateentered in Table 3.
Similarly we apply our local displacement-fitting method to Problem 2. then our
stress-fitting and displacement-fitting methods in turn to Problems 3 and 4. The
resulting estimates are entered in Table 3.
"P. TABLE 3: STRESS INTENSITY VALUES, K1 2
Problem Basis of local Local fitting Actual exactfitting method estimate of K value ofK
1 Stress 0 K
2 Displacement 0 K
3 Stress K 0
4 Displacement K 0
Tabl 3 how tht i ispossible to devise problems wich ensure the
comprehensive failure of the two local fitting methods put forward here.
9
Several comments are in order at this point. Clearly, given knowledge of the exact
solution as here, it is possible to modify either of the two local fitting methods so ,.'
that they do work. In practice, however, one does not know the answer in an actual
application, so that one does not have the option of fixing up a local fitting
approach but can only see if things "look right". In this connection, consider the F
plot 2"ir a(00) for Problem 1, which as it happens coincides with -pu 6 (e 7 7T[)r2'r.
I 2(1 - V) for Problem 2 (Fig. 3): Certainly here things "look right". Indeed one could ,.,
even imagine obtaining results from a sequence of finite element grids at the points
3, 5, 7. 9, then at 2. 3, 4, 5, 6. , .8, 9. and ultimately at 3/2, 2, 512. 7/2, 4, 9/2, 5,
1112, 6. 13!2, 7. 15'2, 8, 17/2, 9. and still concluding K, = 0 when, in actuality, Ki
K. And it is possible to construct a problem which has both its stress fit and its
displacement fit do this. In sum, once any local fitting procedure has been tested
and decided upon, it is then possible to set up an "application" on which the
procedure gives completely incorrect results. Finally we remark that while these
given problems are not the easiest to solve without hindsight using other methods,
they are by no means impossible. In fact, using the superposition method of [3] inconjunction with the path independent integrals of [4] we would judge that
reasonable results could be obtained in return for some computational effort. More *"
precisely, for Problem 1 for instance, we estimate from the participation of the ... ,-*1.
regular eigenfunctions in Table 2 relative to that in trial problems in [4], that K, A.
could be found to within 10% using a uniform grid comprised of around 1000
constant-strain triangle elements, to within 1% using grids with up to 5000 such
elements and extrapolation. Moreover, these analyses could be performed in a
completely systematic way without drawing on a knowledge of the exact answer at
all.
REFERENCES
1. G.B. Sinclair. "The inherent unreliability of local fitting methods forcalculating stress Intensity factors," Report SM 82-19, Department ofMechanical Engineering, Carnegie-Mellon University (1982).
2. G.8. Sinclair. "Some inherently unreliable practices in present day fracturemechanics," International Journal of Fracture, to appear.
3. G.8. Sinclair and D. Mullan, "A simple yet accurate finite elementprocedure for computing stress intensity factors," International Journal forNumerical Methods in Engineering, Vol. 18, pp.1587-1600 (1982).
4. G.B. Sinclair, M. Okajima and J.H. Griffin, "Path independent integrals forcomputing stress intensity factors at sharp notches in elastic plates".International Journal for Numerical Methods in Engineering, Vol. 20 (1984).
--. :...::..
L- 9 C10% LI 7 L
1009S
8 ~~
7--
7
4S
6 M
* 0j
3-0
*0 1/9 2/9 3/9 4/9 5/9 6/9 7/9 8/9 1
r/R
Figure 3. Stress ahead of the crack in Problem I
- - ... \*.~.
~dd
-- V..-.
:~y. -
~I.
iAPPENDIX 4: ON ThE CAPRICIOUS NATURE OF ELASTIC SINGULARITIES I THE '
ELASTIC ANALYSIS OF ThREE RE-ENTRANT CORNERS.'... *~:
'V
.- -.
[
I'
"S
5'-5' .5'
p'S.,
-'5' "S. .5'.
I \*'S*V* 5'."
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S'S...............................
,a-¢
ON THE CAPRICIOUS NATURE OF ELASTIC SINGULARITIES
lo" *
AE Chambers and GB Sinclair
Department of Mechanical Engineering. Carnegie-Mellon UniversityPittsburgh, PA 15213 ,
Classical elastic analysis of re-entrant corners or sharp notches in plates reveals the
presence of stress singularities at the notch vertices As a result, one must make
inferences concerning the structural integrity of components containing such features when
the stresses become infinite in response to infinitesimal loads, clearly not a simple task At
this time the accepted methodology, for the case of cracks in brittle materials anyway, is
to take the coefficient of the singularity - the stress intensity factor - as the parameter
governing failure. This choice represents the basic tenet of linear elastic fracture
mechanics today.
The usual approach taken in the elastic asymptotic Analysis of re-entrant corners is to
examine what stress fields, including singular fields. are possible for a given local geometry
*ere we adopt an inverse approach of prescribing a field with bounded but nontrivial
stresses at the vertex then asking what problem can such fields occur in In this way we
construct a closed form solution with f Mte stresses for a loaded plate in the shape of a
pac-man with a specific angle between its stress-free flanks. This configuration then has .
a stress concentration factor but no stress intensity factor. By maintaining the same
loading on the outer edges of the pac-mar yet perturbing the shape of th flanks about
their original positions, problems which have participating singularities are generated these
last do not readily admit to the determination of close form solutions but are Amenable to
numerical Analysis of demonstrably high resolution using appropriate path independent
integrals and a superposition procedure Since an identical loading acts in all problems and
only very minor changes in geometry occur, it is to be expected that the physirc-
responses would be quite similar. Thus the stress intensity factor's on-off-on-gain type
behavior makes it difficult to envisage how it could be the controlling damage parameter m
this sequence of problems These results. and other like them e.g., references). suggest
the need for a reconsideration of the role of the stress intensity factor in fractre
mechanics.
References
AE Chambers and G.B Sinclair, On obtaining fracture toughness values from 90 tgltrat.r .
Int. J. Fraclure 30, RII-R15 (1986)
GB. Sinclair and AE. Chambers. Strength size effects and fracture mechaics what does
the physical evidence say? En9ng Fracture Mech. to appear.
%.- ,N
* -,r~~r r r r , * r-w-r~~,-i-'.c-cww.~. ~r--~ -7 -~'-~..r*- '-'''~~''*~-
THE ELASTIC ANALYSIS OF THREE RE-ENTRANT CORNERS
A E. Criambers and G. B. Sinclair
SM 86-10 June, 1986
.%
S Department of Mechanical EngineeringCarnegie Institute of Technology
Carnegie-Mellon UniversityPittsburgh, Pennsylvania 15213
L~~ -%p .2.
_1
THE ELASTIC ANALYSIS OF THREE RE-ENTRANT CORNERS ,
INTRODUCTION"-
V..°" °
Since many engineering structures are designed with what are known as re-entrant -.- '-
V.°
corners or notches (Figure 1), it is important to understand the mechanical behavior of " .'
these geometries upon loading. Elastic analysis of these notches can result in stress fields
near the notch tip that are singular in nature [1]. Physically, these singularities do not "-.'
make any sense. Unbounded stresses imply failure under even an infinitesimal load, which .'-.
is inconsistent with everyday experience. For analytical purposes, it is desirable to"-'%
determine the participation of these singular fields in comparison to the regular fields .
present'
These singular stress fields are eigenfunctions of the equilibrium problem posed upon 1
loading re-entrant configurations. In general, most regular eigenfunctions stresses acting
along the bisector of the re-entrant corner are equal to zero, leaving only the singular
fields to dictate the mechanical behavior of the notch (Figure 1). It is thesn stress fields
acting at h te notch tip that are analytically unbounded. I is also these stress fields tat
require making some sense of their significance in the problem under investigatio In
order to gain some understanding as to how the singularity plays a role in the overall .,,,
stress field, one could ask the question, is there a re-entrant geometry which has non-,-.-.
zero and finite stresses acting et te notch tip Such a configuration would have no
singularity participation, Le. a stress intensity factor of zero. The answer to this question is, ,,,
in fact yes. Upon finding re-entrant corner that has only regular stress fields, the
-. present-
objective of the present work is to track the variations of the stress intensity factor with
alight perturbations of geometry. A re-entrant corner under a particular loading that has a
stress intensity factor of zero yet nontrivial stresses t thi vertex will be determined. The
angle near the vertex of the corner will be decreased and then increased slightly The
raeecedi
-oUo°
orde to ain omeundestaningas t howthesinglariy pays rol intheoveral U
,-o-,,, ,*,•. ,. stes fel , ne could ask -. th qesi,. is thee.a.reentran geme .which has-..', non- .. . ... ,
% r e.0 *
'I
lot*t
Figore j
3
effects of these geometric changes will be studied through the behavior of the stress "
intensity factor.
The first section presents the formulation of the problem under investigation. The
exact geometries to be examined are introduced. The generalized stress intensity factor is \;"
defined. The second section presents the numerical analysis. A finite element approach is
taken to determine the stress and displacement fields. A path-independent integral is used
to calculate the stress intensity factor with the information produced from the finite .0
element analysis. Results will be presented and discussed in the final section
PROBLEM FORMULATION
In order to study the effects of a stress singularity of a re-entrant corner, three
individual regions are analyzed. The basic difference of the three regions is that the
material angle local to the notch tip is slightly perturbed from an initially symmetric.-.
geometry. The planar region R has material angle, 2a (i=0,1,2). In particular, a0 =
128.727 = 125.863 and a 2 = 131.863. The choice of these particular values will
be explained later. Let (x,yl be cartesian coordinates that have origin at the notch vertex,
and 'ie x-axis bisects the material angle there (Figure 2). Let (r.6) be polar coordinates in
the region where r2 x2 + y and 8 = arctan (y/x). The dimensions of the notch are as
follows: a = notch depth and h = half-height of notch.
The region, R is defined as the region enclosed by basically two boundaries. The
first boundary, a 1Ro. is the boundary to which the prescribed tractions will be applied. It
consists of three sides of a rectangle. The line segments that define it are:
y = ±h for -a<x< 2h and x = 2h for -hSygh (1)
The second boundary, a 2R0, to enclose the region, R0, is the two line segments given by
tana ±y/x for -(x. 12)
The region, RV, is defined in a similar manner. The local x,y) cceo-dinate system near ,=:"
the vertex of the corner is slightly rotated about the global coordinates by about Z = 3*.
The outer rectangular boundary, O lR1, on which the tractions are applied is left intact, as
~-~- ,-w"-~- ~~YVV'~V~ J . ~V -~ 7 '~V' ~ d F ~ 7.' 1 ~ ~ . F
4
* .*~ *p.
.1
iw~ i-I
0
L" I
Id**
h HI ..4bC I
r-j
- p-I
4
FI~LPCe -.01
.1*
.~~~~ V.'' . "WV€W 7
.J. . ..
5
is the lower flank. The upper flank is rotated by approximately 26 and extended to y =
h/2, thus resulting in less solid material to support the applied loads in the original problem.
It is connected by a smooth curve to the point where the positive y-face boundary begins be%
in the second quadrant at y = h. This is boundary a 2 R1.,
The perturbation of the upper flank for Problem i = 2 is similar to that in the first
perturbed problem. However, the rotation of the local axes is in the opposite direction,
resulting in more solid material than both Problems i = 0 and 1. This is region R2. (The
dotted lines in Figure 2 represent the upper flanks for Problems i = 1 and 2.)
Formally, we seek the stresses, orr " and r and the displacements, u6 and u,
as funclions of r and 8 in the region R, satisfying the following. The stress field must
satisfy the two- dimensional stress equations of equilibrium in the region in the absence of -
body forces,
r+ - o + rr 60 0
ar roc)G9 r 4
6 + -- + = 0, (3)r r ar r
on R. The stress-displacement relations for a homogeneous, isotropic, linear elastic plane
hold for the regions.
2prulu 1rr au I ar
2p (1 u\+ (4)
ii..- (I*
r au +auO uO7 r9 =rO rae3 ao r '
• , - " -
6
on R, where p is the shear modulus, C1 equals 1-2v for plane strain and 1-v for plane
stress, C2 equals 1-v for plane strain and 1 for plane stress, and v is Poisson's ratio.
The notch flanks are stress-free which requires that
O0 0 r =0 on (2 R when ± a.. (5) %
2 ii
The remaining boundary conditions on aoR, are prescribed tractions derived from the
following stresses:
I rrJ• -I - ~Orr -0 0 sin26 + - sin2 0 0
ao
=0 sin2 - sin200 (6)88 a 0 .:... ,
7 = - cos20 - cos20 0 ] -
where '0 is arbitrary. Finally, the regularity requirements involve the fact that the0'
displacements at the vertex of the notch must be finite,
u and u,9 = 0(0) on R. as r-O. (7)
More specifically, we want to use the stress and displacement fields found in the
problems (i = 0, 1 and 2) formulated above to find the generalized stress intensity factors ,.,
present There are two possible factors, one arising from any symmetric loading and one ...
arising from any antisymmetric loading in the problem. For the symmetric case, the stress
intensity factor is defined as:
K1 =_ fn, r 1-, on 6 0. (8)
The subscript I represents mode I loading (symmetric opening mode). The value of X is
the eigenvalue identified after Williams [1] that represents the only singular stress field
possible at the re-entrant corner under this symmetric mode. That is, X. is the root of the
eigen-condition
sin2A a =-Xsin2a. IN . .1). (9)I I I I I .
For the anti-symmetric case, the stress intensity factor is defined as
. . . . . . , . "
",p ",j "J
°.,° ..
'S.
7
Kt V + ,g r're on e=o, (10)
where the subscript II represents the mode II loading (antisymmetric). The eigenvalue X, is -,
evaluated from the antisymmetric eigen-condition,
sin2Xa = ), sin2a. (0<X,<1). 11), , I.. .-
4% These eigenvalues produce eigenfunctions which are the only singular stress fields possible
for the particular notch geometry. As a result of the problem formulation, the
displacement fields derived are regular. That is, the fields near the notch tip behave as
Gee and Tr =O(ri ) and u and u. = O(r i) on R as R--0. (12)
Some explanation of the traction boundary conditions found from (6) is now in order.
Recall, that the purpose of this study is to look at slight geometric perturbations of a
regular configuration. Therefore, R0 is taken to be the region in the nonsingular problem
under investigation The approach taken is such that a particular material angle, a0 is
sought that will produce a regular and non-zero stress field for an antisymmetric problem. -'-
The following transcendental equation results:
tan2a 0 = 2( (13)0 0
It is in solving this equation that a is found to be approximately 128.70 The remaining
two problems are then posed by simply changing the material angle by plus or minus about .S
three degrees. Note, also, there is now an exact solution for the nonsingular problem that
is independent of the radial coordinate, r. It is these regular stress fields that are used as
the traction conditions on the boundary a 1R (i=0,1,21.
-'S
In conclusion, what we have are three problems where the applied tractions and the
boundary on which they are applied remain the same. The flanks change in geometry, but
remain stress free. An easy way to imagine it all, is that the region R0 is an ideal
geometry of a re-entrant corner that is to be fabricated (i=0). The first singular case
examined (i= 1) is a result of machining just a little too much material away from the upper
flank. The second (i=2) follows from not machining quite enough material away from the
.V V'i' " - '" S " . .~ - - - . -.. -,
8
.
flank. When the three configurations are loaded under the same conditions, what is the
result of these slight geometric changes?
ANALYSIS .
The analysis of the problems requires two major steps. First, the stress fields in R (i .,'5'
= 0,1,2 ) need to be determined. Secondly, as stated in the problem formulation section, a
method is then needed to determine the stress intensity factor of each problem for both
mode I and mode II loadings.
A finite element method is used to numerically determine the resultant stress and
displacement fields of the three problems.* For each problem, three grids of different
sizes are used in the hopes of attaining better accuracy and gauging convergence. The
grid refinement is fairly systematic. The coarse grid for the nonsingular problem ( i0 ) is
shown in Figure 3. The nonsingular stress fields defined in (6) are applied on the
- rectangular boundary, leaving the notch flanks stress-free. Upon finite element analysis,
"" nodes along common e are of constant stress. (One drawback of the grid pattern chosen
is that near the notch vertex there is no refinement in the angular direction. Therefore,
convergence as r goes to zero is difficult to judge, though, divergence can be obvious.)
The coarse grid for the first singular problem (i=1) is shown in Figure 4, while that for the
second singular problem is displayed in Figure 5. The dotted lines in Figures 4 and 5 are
the smooth curves that represent the flanks. Since the finite element code uses linear
elements, these curves are better approximated upon grid refinement For all three
problems, three grids were used in the stress analysis. The coarse grids have 68 elements
and 48 nodes. The medium grids have 264 elements and 159 nodes. The fine grids have
1040 elements and 573 nodes..5,-.
The second step of the analysis requires a method of determining the generalized
stress intensity factor from the stress and displacement fields generated from the finite
element analysis. The method employed, the H-integral 12], is from the family of path-
independent integrals. The major advantage of path-independent integrals is that the only
The prograrn used is PLANDJ, originally developed by J.L. Swedlow. It is used on the VMS/VAX at the .. ,'.
- Mechanical Engineering Department. Carnegie-Mellon University.
• .. .4 %
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12
errors accumulated are those from the numerical approximations present Another
advantage of these integrals is the fact that the symmetric and antisymmetric parts need
not be separated in the analysis. (For the argument supporting this statement, see
reference [3).) The path of integration is shown in Figures 3, 4 and 5. Notice that the
inside path is chosen so that the curvature of the notch flanks in the two singular
problems (i = 1 and 2) does not interfere with the independence of the integration path.
The H-integral program uses the second-order trapezoidal rule for the integration. The
nodal values along the path of integration are used as points of numerical integration. The
complementary stress fields and displacement fields used in the H-integral (see reference
[2)) are determined from the (r,A) coordinates of the nodes along the path. The actual
nodal displacements are read from the finite element output The stresses at the nodes are
determined from nodal averages of the surrounding elements of each node along the
integration path.
RESULTS
In presenting the results of this analysis, some sort of verification is required. One
common method of verification of the finite element method is to check convergence at
nodes common to all three grids used for each problem In the nonsingular case,
convergence to the exact solution (6) is required. In the other two cases, convergence
within the three associated grids is desired. As stated earlier, since there is no grid
refinement in the angular direction, convergence near the origin is difficult to assess.
Divergence, on the other hand, is clear when present Therefore, the nodal stresses acting
along the x-axis (0=0) are plotted for all three problems. (Figure 6, 7, and 8.)
Convergence to the exact solution in the nonsingular problem (i=0) is seen clearly (Figure 6).
The antisymmetry in the problem is observed in that or. = 0 and -r. is finite along the
x-axis. Convergence at r=0 for r is not obvious (though, as explained earlier, it is not
particularly expected). However, divergence is not noticed.
Comparing the expected singular problems (i = 1 and 2), divergence at the origin is
clear. For the singular problem i=1 (Figure 7), ar80 at r = 0 is divergent, where the trend
of 7r8 is not obvious. This is consistent with what will be seen later, problem i = 1 has a
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mode I singularity and no mode II. The results for a,9 9 and TrI for problem i=2 (Figure 8) .
are divergent at r=O This is also consistent since both mode I and 11 singularities are
present in the problem.
Another method of verification is the fact that where there is no singularity present,
H : K should be zero. The H -integral of the nonsingular problem oscillates about zero.
And is. in fact, numerically zero for our purposes. (The H -integral cannot be determined .*
for the e gervalue equal to 1.0, see reference [2).)
For the nonsingular problem (i=0), the eigenvalues corresponding to both equations (9)
'- and (11) are simply equal to 1.0. Therefore, the stress and displacement fields are of the
* order one The mode I eigenvalue of the first singular problem (i=1) is X 0.5818: this
is the only singular stress field in this problem. For the second singular problem (i=2),
however, singularities in both modes exist That is, X = 0.5554 and X = 0.9528. The
corresponding results of the H-integration are presented in Table I.
Notice how the value of the dimensionless K in problems i= 1 and 2 are virtually equal
in magnitude but opposite in sign. This could have been expected from the beginning since
the original (nonsingular) material angle is perturbed about the same in both directions for
each of the singular problems (i.e. a,1 = a- 6 and a2 =a +). Notice, also, that the value
of KII for problem i=2 is larger in magnitude than K. This is expected since the loading
used was derived from an antisymmetric problem. The antisymmetric singularity should then
be dominant
CONCLUDING REMARKS
In general, re-entrant corners produce stress fields which are singular as the notch
vertex is approached. However, a configuration can be found at a particular material angle
and loading that will produce all regular stress fields. Slight perturbations in geometry
produce predictable behavior in the stress intensity factor. Since the applied loading in this
particular problem is based on an antisymmetric configuration, the mode II singularity has a
larger participation factor than that of the mode I. Perturbing the material angle in equal
magnitude but opposite direction, results in equal but opposite stress intensity factors for
mode 1.
_ 4
17 " °
Table t. Dimensionless Stress Intensity Factors (K = K I (ul-J a-X ) "."p
-aKrModem/ Coarse Medium Fine
Problem 0Mode I 1.000 128.70 0.000 0.000 0.000
Problem 1Mode I 0.582 12590 -0.066 -0060 -0058
Problem 2 .,Mode I 0.555 131.90 0.063 0.060 0,059 ",-Mode II 0.953 131.9 ° 0.648 0.610 0632
= eigo-value after Williams 1 .,-
= half the material angle of re-entrant corner
Even though the behavior of the stress intensity factor may be predictable, the
problem still exists in that these singular fields are not able to interact with the other
regular stress fields that are present The divergent stresses acting along y = 0 in the
singular problems (i = 1 and 2) highlight 1;is difficulty. Studying the behavior of the stress
intensity factor may lead to a better understanding of what is occuring :n these re-entrant
configurations.
References
1. ML. Williams, "Stress Singularities Resulting from Various Boundary Conditions in
Angular Corners of Plates in Extension." Journal of Applied Mechanics 19 (1952) 526.
2. GB Sinclair, M. Okajima and J.H. Griffin, "Path Independent Integrals for Computing
Stress Intensity Factors at Sharp Notches in Elastic Plates," International Journal for
Numerical Methods in Engineering 20 (1984) 999.
3. G.B. Sinclair, "A Remark on the Determination of Mode I and Mode II Stress Intensity%..p
Factors for Sharp Re-entrant Corners," ,International Journal of Fracture 27 (19851 R81.
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pAPPENDIX 5: ON OBTAINING FRACTURE TOUGHNESS VALUES FROM THE LITERATURE
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International Journal of Fracture 30(1986) R I -RIS ., ..0 Martinus Nijhoff Publishers, Dordrecht - Printed in the Netherlands
ON OBTAINING FRACTURE TOUGHNESS VALUES FROM THE LITERATURE
A.E. Chambers and G.E. SincZairDepartment of MechanicaZ Engineering, Carnegie-Mellon University .,Pittsburgh, PennsyZvania 15213 USAtet: (412) 578-2504
Recently, extensive compendiums of sources of fracture toughness datahave been assembled by Hudson and Seward [1,2]. These lists of referencesnaturally prompt the question as to what an engineer would find upon con-sulting them in terms of actual values of fracture toughness for a givenmaterial. The intent of this note is to indicate an answer.
In choosing specific metals for which to obtain fracture toughnessvalues, we look to ones with large listings of data sources so as togauge any variability present. To this end we select AISI 4340 steel and7075-T6 aluminum. We focus on plane strain fracture toughness values(K Ic ) as governed by ASTM standards since these are more generally regard-ed as being material properties. We do not check, however, whether or nota given test furnishing a K value complies with all of the specificationsin ASTM E399 [3], simply belCuse none of the references reviewed provided ISsufficient information to enable a complete check. Hence values are in-cluded as valid K. if their contributors claim them as such. Since wefrequently encountered plane stress fracture toughness values (K ) in the
cdata for 7075-T6 aluminum, we include these as a separate set for com- -.. .parison. Further by way of comparison, we note yield strengths (a),since a may reasonably be viewed as the uniaxial tension test qua[tityanalogogs to initial unstable crack propagation in a brittle material.In processing the data we distinguish between markedly different specimen % .types reported in a single source but otherwise use mean values for each -.-k".'source. That is, when what is in essence the same test is repeated a .number of times and outcomes recorded, we merely extract the mean*. Forthese average values, we note the number of essentially independent sourcesand calculate an overall mean. To estimate the variability we also cal-culate the ranges and 95 percent confidence intervals (1.96s, s being thestandard deviation). In justification of the' second, histograms of thedata show good agreement with the expected frequencies in a normal distri-bution (possibly because the individual data typically represent meanvalues themselves and the central limit theorem applies to some extent).The only exception to this agreement occurs for the K, values for 7075-T6.These data, though, conformed well with a normal distfibution on takinglogs and accordingly a log transformation was used to determine the confi-dence interval in this instance. The results are sa rized in Table 1;details of K values are shown in Tables 2,3 wherein single numbers inbrackets denoie original sources, hyphenated numbers the correspondingreference in [1] or [2], and a virgule between the two implies as reported.in the latter, e.g. [61/11-12] is [6]'s data as drawn from [12] in [1]**.*For the 7075-T6 KI , about a quarter of the results were designated asbeing for a longituSinal orientation and a like fraction as being for atransverse orientation with the remainder being unspecified. Given thatthe difference between the means for all the longitudinal and transversecases was found to be less than 2.5 percent, the effects of orientationfor this alloy were not regarded as being sufficiently significant to meritdistinct classification.**In the interests of brevity we do not relist references given in [1], [2].
Int Journ of Fracture 30 (1986)
op'- ... •..
R12
Examining Table 1, we find that the variability in plane strain frac-ture toughness for AISI 4340 steel is 100 percent of the overall mean whenbased on the range, 87 percent when based on the confidence interval Thecorresponding percentages for 7075-T6 aluminum are 162 percent and 108percent. These wide fluctuations are in marked contrast to the respectivepercentages for the yield strengths which are 17 percent and 19 percentfor the steel, and 22 percent and 21 percent for the aluminum. Indeedthey are quite comparable to the variations in plane stress fracturetoughness values.for 7075-T6, viz, 110 percent and 72 percent*.
Such diverse It data demonstrate that the engineer needs to exerciseconsiderable care 19drawing a value of plane strain fracture toughness .
for a particular material from the literature. Short of undertaking the .1time consuming task of collecting a sufficient set of references as a way ZS-of assessing variations as here, no reasonably certain methodology forensuring a conservative estimate of fracture toughness appears to exist.Possibly, if a single value is found and then assumed to be at the upperlimit of the ranges involved and a safety factor applied to reduce it tothe lower, a conservative KT could be expected. Such safety factorshere would be 2.9, 2.5 for ESI 4340 and 3.7, 2.9 for 7075-T6. Thesesuggest that a safety factor of 3 might be adequate, although there isreally no guarantee of this being so for another material. Indeed, thefact that the plane strain fracture toughness data display more than five"
times greater variability than the yield stress data and vary about as %
such as the plane stress values which are known to be geometry dependent,raises serious doubts concerning the notion of Kic being a material pro-perty. An engineer continuing to interpret and use it as such may there-fore be making significant errors. NA
Acknowledgement: The financial support of this investigation by the 14
Air Force Office of Scientific Research is gratefully acknowledged.
REFERENCES
[l] C.M. Hudson and S.K. Seward, International Journal of Fracture 14
(1978) R151-R184. .."-
[2] C.M. Hudson and S.K. Seward, International JournaZ of Fracture 20(1982) R59-R117.
(3] E399-83 Standard Test Method for Plane-strain Fracture Toughness ofMetallic Materials, 1985 Annual Book of ASTM Standords, Volume 3.01,American Society for Testing and Materials, Philadelphia (1985) 547-582.
[4] W.F. Brown, Jr., and J.E. Srawley, Plane Strain Crack Toughness Teat- 6---g
ing of High Strength Metal lic tNeriaZe, STP 410, American Society ofTesting and Materials, Philadelphia (1967).
[5] J.E. Srawley, M.H. Jones, and W.F. Brown, Jr., Materials Research andStandards 7 (1967) 262-266.
[6] E.A. Steigerwald, Report AFML-TF-67-187, Air Force Materials Labora-tory, Wright-Patterson AFB, Ohio (1967).
[7) B. Mravic and J.H. Smith, Report AFML-TR-71-213, Air Force Materials e-,Laboratory, Wright-Paterson AFB, Ohio (1971). %
*These plane stress percentages are similar to, though somewhat lower than, .' .
the corresponding numbers for 2024-T3 aluminum gathered from nineteen Ne*.
independent sources.
Int Jrourn of Fracture 30 (1986)
X4
I "..".
R13-
[8] M.F. Amateau andE.A. Steigerwald, Report AD611873 (1965).191 S.lH. Jones and W.F. Brown, Jr., in Review of Developments in Plane @_
Strain Fracture Toughness Testing, STP 463, American Society forTesting and Materials, Philadelphia (1970) 63-91.
[10] R.H. Heyer and D.E. McCabe, in Review of Developments in Plane StrainFracture Toughness Testing, STP 463, American Society for Testing andMaterials, Philadelphia (1970) 22-41.
[111 U.H. Lindborg and B.L. Averbach, Acta MetaZZurgica 14 (1966) 1583-1593.
[12] G. Sachs, V. Weiss, and E.P. Kiler, Proceedings of the American So-ciety for Testing and Materials 56 (1956) 555-565.
[13] G. Succop, M.H. Jones, and W.F. Brown, Jr., "Effect of Some Testing -'.Variables on the Results from Slow Bend Pre-cracked Charpy Tests",NASA Lewis Research Laboratories, Ohio (1975).
[14] T.M.F. Ronald, unpublished data, Air Force Materials Laboratory,
Wright-Patterson AFB, Ohio (1974).
[15] W.S. Pellini and J.E. Srawley, NRL Report No. 5609 (1961).
[16] D.Y. Wang and D.E. McCabe, in Mechanics of Crack Growth,. STP 590, * --
American Society for Testing and Materials, Philadelphia (1976)169-193.
[17] D.A. Gerdeman, J.C. Wurst, J.A. Cherry, and W.E. Berner, ReportAFML-TR-68-53, Air Force Materials Laboratory, Wright-Patterson AFB,Ohio (1968).
[18] A.R. Simpson and J. Fielding, BSA Report ER/MISV/MAN/1050 (1971).
[19] J.G. Kaufman and M. lolt, Technical Paper No. 18, ALCOA Research Lab-oratories, Pittsburg, Pennsylvania (1965).
[20] H.P. van Leeuwan et-al., NLR Report Nos. TR 72032, TR 72082, TR 72088,Amsterdam, The Netherlands (1972).
[21] G.T. Schaeffer and V. Weiss, Interim Technical Report No. SUI MET-1078-664-12, Syracuse University, New York (1964).
[22] R.J. Goode, unpublished data (1974).
[23] D.W. Hoeppner, in Case Studies in Fracture Mechanics, Report No.AMMRC MS 77-5, Army Materials and Mechanics Research Center, Mass.(1977) 3.1.1-3.1.10.
[24] S.O. Davis, N.G. Tupper, and R.M. Niemi, Report AFML-TR-65-150, AirForce Materials Laboratory, Wright-Patterson AFB, Ohio (1965).
[25] W.F. Brown, Jr. and J.E. Sravley, Data presented at Subcommittee IIIMeeting of ASTh Committee E-24 (1964).
[26] W.F. Brown, Jr., "Progress Report on NASA-NRL Cboperative PlaneStrain Testing Program", ASTM Committee E-24 Meeting (1965). WAN
[27] M.V. Hyatt and M.O. Spidel, in Stress-Corrosion Cracking in HighStrength Steels and in Titaniwn and Aluri nw Alloys, Naval ResearchLaboratory, Washington, DC (1972) 147-244.
[28] R.E. Jones, Report AFML-TR-69-58, Air Force Materials Laboratory,Wright-Patterson AFB, Ohio (1969).
[29] F.C. Allen, in Deage Tolerance in Aircraft Structures, STP 486,
Int Journ of Fracture 30 (1986)
49--A.
......................% N -. . .
R14 'er
Amserican Society for Testing and Materials, Philadelphia (1971) 16-.iw38. ~?
3 December 1985
Table 1. Mean valuen and variability of some material data
95% *~
Material Quantity No. of Mean Range Confidence
data value interval
Steel- a 36 1500 1370-1630 1360-1650
AISI 4340 K1 32 70.5 37.4-108.2 39.8-101.2
Aluminum- CY 42 507 434-545 452-561-y
7075-T6 K I 29 34.5 21.0-76.9 19.4-56.7%
K c94 70.7 32.9-111.0 45.1-96.2
1/2Note: W values are in M~a; Kic K values are in M~am
y Icc
Table 2. Plane strain fracture toughness values for AISI4340 steel4
K-1 c 1 KIc Ic11(Ma m )Source (Mpa m ) Source (MPa m ) Source
64.8 [1-21 70.8 [11/[1-15] 108.2 [121/[1-1561
57.7 [4]1[51/ 53.8 [1-25] 57.7 [1-160] S. .1
76.9 [1-121] 3. 58.2 [131/[2-31
58.2 [6]/i[-12] 76.989.0 37.4 [1-1521 89.090.1 65.45.2 [4/23
65.9
747 7/[-1] 65.9 57.6 12-271
85.7 [83/fl-iS] 67.0378 [5[210
74.7 [9]I[1-15] 7. 63.0
84.562.6 [2-329]87.9 [10]/[1-151 87.9
rmt Journ of Fracture 30 (1986)
.11j -
R15
Table 3. Plane strain fracture toughness values for 7075-T6 aluminum
* K K K .
1 1 /2 ) Ic1/2 c1/2)(MPa M Source (MPa m ) Source M~a m Source
37.0 [161/1-2] 31.6 [20]/[1-51] 26.6 [2-66]
21.0 [1-5] 30.8 [1-52] 28.2 [241/[2-130]
27.7 24.2 [2-2] 31.9 [25]/[2-130J
38.5 [1-6] 39.1 [21]/[2-3] 29.9 [26]/[2-1301
48.9 25.3 [22]/[2-3] 26.4 [27]/[2-137]
35.9 [1-11] 34.5 [231/2-27] 35.2 [2-153]
48.1 76.9 21.6 [2-1571
53.7 34.8 [2-39] 32.6 [28]/[2-1591
29.1 [171/[1-12] 35.7 [2-59] 34.8 [291/[2-1591
28.6 [18]/11-15]
30.8 [191/[1-24]
% a
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